# Building the sense of math in physics the...1 Building the sense of math in physics activities Physics 131, Fall 2016, E. F. Redish Quiz questions Worms get oxygen for their metabolism by absorbing it through their skin. The oxygen is then circulated throughout the worm's ...

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Building the sense of math in physics activities Physics 131, Fall 2016, E. F. Redish

Quiz questions

Worms get oxygen for their metabolism by absorbing it through their skin. The oxygen is then circulated throughout the worm's volume by a circulatory system with multiple hearts. In order to understand how the physical mechanism of absorption and the parameters that determine the rate of absorption of oxygen per unit area (A) and the rate of use of oxygen per unit volume (B) constrain the worm's size, we'll make a simple model of the worm: a cylinder with flat ends as shown in the figure.

In this model of the worm, its size and shape are determined by two shape variables: its length, L, and its radius, R. For our first problem, we will ignore the effect of the end caps (the flat circles on the ends of the cylinder).

IV. The graphs at the right show the absorption and use of oxygen for the whole worm as a function of its shape variables. Four graphs are shown, labeled by the Greek letters alpha (α), beta (β), gamma (γ), and delta (δ). Choose the Greek letter that identifies the curve that is most likely to represent:

a) The total amount of oxygen absorbed by the worm as a function of its length (L).

b) The total amount of oxygen used by the worm as a function of its radius ®.

c) The total amount of oxygen absorbed by the worm as a function of its radius ®.

The model that ignores the effect of the ends is only good if the radius is much smaller than the length.

2. If we include the areas of the flat circles on the ends of the cylinder, we get a correction to the total area. Complete the following sentence by putting the number that belongs in the blank space:

"The correction to the area of the worm in our cylindrical model due to the endcaps will be less than 10% if L __ R."

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1. The diffusion constant D, describes how molecules jiggling around in a fluid drift away from their starting point. It has dimensions [D] = L2/T. We have good reason to believe (we’ll see it in a reading later) that in a dilute gas, D only depends on the average distance a molecule travels between collisions, λ, and it’s average speed v. If [λ] = L and [v] = L/T find an equation that expresses D in terms of λ and v.

2. Given that one of the following correctly represents the surface area of a sphere of radius R, can you tell which equation it is? If so, choose the correct letter. Otherwise choose f).

a) 4

3 πR3 d) 4πR3

b) πR2 e) 2πR c) 4πR2 f) I do not have enough information to tell.

3. Given that the average size of a human cell is about 10 microns across, which of the following provides the most reasonable estimate for the number of skin cells in your hand?

A. 100 F. 108 B. 102 G. 1010 C. 104 H. 1012 D. 105 I. 1014 E. 106 J. More

4. You fly east in an airplane for 100 km. You then turn left 60 degrees and fly 200 km. About how far are you from your starting point?

A. 170 km B. 200 km C. 260 km D. 300 km E. 370 km

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1. The Moving Man walks steadily towards the tree for 6 seconds, then stands still for 6 seconds, and then towards the house twice as fast as before for 6 seconds. Which pair of graphs would the program show for his position and velocity?

2. Consider a model of predator and prey in which the predator and prey detect each other suddenly at the same location. The prey would have to accelerate faster than the predator in order to escape. The predator would accelerate more slowly than the prey (it's larger) but attain a higher velocity, as in the graph shown. If the prey has ventured 4 seconds away from its burrow and safety, and if the graphs are a good model for the velocities of the predator and prey, in this case will the predator catch the prey?

3. A torus is the mathematical name for the shape of a donut or bagel. Its volume can be expressed as a function of the inner and outer radii of the torus (distance from the center to the inner and outer edge). Which of the following equations could be the correct equation for the volume of the torus?

A. V = 2πR( ) πr2( ) B. V = π

2

4 R + r( ) R − r( )2

C. V = πRr

D. V = π 2

4 Rr( )2

4. True or false? The velocity dx/dt can only be negative if x is negative.

3. One of the essential elements of the animal immune system is the macrophage: a cell that ingests potentially harmful bacteria and destroys them. The bacterium might contain molecules of a chemical safe for it, but harmful to the macrophage. But the macrophage is bigger, so the density of the harmful

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molecules will be less. (Assume they are both in a fluid and approximately spherical.) Suppose the macrophage has a diameter of 20 μm and the bacterium has a diameter of 1 μm. If the density of these molecules in the bacterium is D, what will be their approximate density in the macrophage once the bacterium has been ingested, broken up, and distributed throughout the macrophage?

A. D B. D/20 C. D/400 D. D/8000 E. Something else

1. Real springs only follow the Hooke's law model for small displacements around their rest length. For a realistic spring like the one shown in the figure at the right, it behaves as follows: for stretching from its rest length, it obeys Hooke's law for a while, then as you stretch it further, it gets stiffer as the coils begin to bend. Eventually it straightens out into a long straight wire which is very hard to stretch at all. If you keep pulling harder, the wire suddenly goes plastic (stretches easily) and breaks. If you try to compress it, the coils almost immediately get pushed together and you can squeeze it very hard without getting much compression. Which of the following graphs correctly represent the best Force vs. Length curve for this spring?

2. The magnitude of the viscous force on a small sphere of radius R moving with a velocity v in a dilute gas is given by the equation Fgas→sphere

viscous = 6πμRv where μ is the viscosity.

2.1 Use dimensional analysis to find the dimensionality of μ (in terms of M, L, and T).

2.2 In the middle of the 19th century, James Maxwell showed (both theoretically and experimentally) that the viscosity of a dilute gas only depended on the mass of a molecule of the gas, m, the average distance a molecule in the gas travels between collisions (mean free path) λ, the density of the gas, n (number of molecules per unit volume), and the average velocity of molecules, u. Use dimensional analysis to find a formula for μ in terms of m, λ, n, and u.

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2. In a cylinder of volume V0, we have 1 mole of NO2 at a pressure p0. Pins hold the piston on top of the cylinder in place so the volume cannot change. A spark in the chamber results in the following reaction:

2NO2(g) N2(g) + 2O2(g).

All of the NO2 molecules break up and the cylinder now contains only N2 and O2 molecules. After some time, the temperature of the gas mixture returns to the original temperature through exchange of heat with the heat sink. At that time,

2.1 Compared to the initial state, the number of molecules that are in the cylinder has

b. Increased c. Decreased d. Remained the same

2.2 Compared to the initial state, the pressure in the cylinder has

a. Increased b. Decreased c. Remained the same

1. In the figure are shown four cases with different arrangements of charge. Each charge has the same magnitude, but some are + and some are -. All distances are to the same scale. We were considering the force exerted on a positive charge placed at the points, P.

1.1 If the positive charge, P, is replaced by a negative one, what happens to the direction of the electric force it feels?

A. The direction of the force stays the same in all the cases. B. The direction of the force reverses in all the cases. C. The force changes magnitude in all the cases. D. The direction of the force only stays the same in some of the cases. E. The direction of the force only reverses in some of the cases F. The force changes magnitude only in some of the cases.

1.2 If the positive charge, P, is replaced by a negative one, what happens to the direction of the electric field it measures?

A. The direction of the field stays the same in all the cases. B. The direction of the field reverses in all the cases. C. The field changes magnitude in all the cases. D. The direction of the field only stays the same in some of the cases. E. The direction of the field only reverses in some of the cases F. The force changes magnitude only in some of the cases.

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2. In the figure at the right, a test charge is placed near a molecule to explore the electric force around the molecule. Which of the following statements are true? Assume each atom is a small charge that does not polarize significantly.

A. The test charge is attracted to the molecule. B. The test charge is repelled from the molecule. C. The test charge does not feel an electric force from the molecule. D. The molecule has a net positive charge. E. The molecule has a net negative charge. F. The molecule is neutral.

1. In the figure at the right are shown four configurations of three charges, all having identical magnitude. The diagrams are to the same scale. Rank order the total electrical potential energy of each configuration and indicate if any of them are zero. (For example: E=0>F>G>H indicates E is 0 and F, G, H are increasingly negative. You use A, B, C, D and the correct ranking. Only use greater than, ">", and equals, "=" signs. Do NOT use "

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Consider two cells that are suspended in solution.

3. Model these cells as two sacks, each made of a semi-elastic membrane surrounding a homogenous colloidal fluid. Suppose we have two cells that are different sizes and attached. If the interface between them is a flat plane (as shown in the figure at the right above), what can you say about the internal pressures of the two cells?

a) They are equal b) The larger cell has a greater internal pressure c) The smaller cell has a greater internal pressure d) This cannot be determined with the information given

Exam Questions

1. In class, we analyzed a video of a juggler catching and throwing a ball. In the frame shown at the right, he has caught the ball and is holding it so the ball is at rest. This corresponds to the time t = 0. He then throws the ball upwards and we watched it reach the top and come most of the way down. By taking position and time data, we were able to have the program find velocity and acceleration for the ball at each time. A smoothed version of the acceleration graph for the range of times described is shown below.

For each of the following physical descriptions of an instant of time, select the letters on the acceleration graph corresponds to those instants. Be sure to give all the instants that most likely match the physical description. If none match, put N.

1.1 At this instant, the ball is moving upward with its maximum speed. 1.2 At this instant, his hand releases the ball. 1.3 At this instant, the ball has velocity 0. 1.4 At this instant, the ball is traveling upward. 1.5 At this instant, he is exerting no force on the ball.

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2. Because of their light weight and flat bottoms, basket coffee filters are good ways of probing the resistive forces exerted by the air on objects moving in it. (A stack of them are shown at the right.) If the filter is falling straight down and smoothly, the inertial drag that the air exerts on it is given by Fair→ filter

inertial drag = 12 CρAv 2 where C is a dimensionless constant, ρ is the density of

the air, A is the area the filter presents to the air (~the area of the bottom circle of the filter), and v is the velocity of the filter moving through the air.

A. As the filter falls, it speeds up until the resistive force pointing up is equal to the force of gravity moving down. It then moves with a constant terminal velocity, vT.

A.1 If the filter has a mass of m, find an equation that expresses the velocity with which the filter is moving once these forces balance. Show your work.

A.2 Suppose you have measured the mass of the filter to be 1 gram, its radius to be 8 cm, and its terminal velocity to be 1 m/s. If you take the density of air to be 1 kg/m3 and the gravitational field to be 10 N/kg, find the value of the constant C.

B. The Reynold's number, Re, for an object moving in a fluid is the inertial drag force given above divided by the viscous force, Ffluid→ filter

viscous = 6πμRv where μ is the viscosity of the fluid, R is the radius of the object and v is its velocity through the fluid. (This is actually correct up to a dimensionless factor. For this problem take Re to be the ratio of these two forces.)

B.1 Write an equation for the Reynolds number for this example, simplifying the equation as much as possible (e.g., cancelling factors that are both in the numerator and denominator).

B.2 If the viscosity of air is about 10-3 kg/m-s, find the value of the Reynolds number for the coffee filter falling in air.

B.3 From your calculation, which force do you expect to dominate for a falling coffee filter?

3. Different cells have many of the same components but they often have different amounts of those elements. Sometimes, those differences are just because one cell is bigger than the other. Other times, it's because the cells have different structures and functions. For three common model cells, three of these elements are membrane proteins, mRNA molecules, and ribosomes. The number in each cell is shown For one (only one!) of these elements, decide whether the difference in number in the three cells is simply because of the different sizes of the cells (approximately – to, say, within a factor of 5 or so) or whether there might be some interesting difference between the cells. (Be sure to tell us which element you are considering!) Be sure to clearly state your assumptions and how you came to the numbers you estimated, since grading on this problem will be mostly based on your reasoning, not on your answer.

4. In this class, we use equations not just to carry out numerical calculations, but also to express information about physical systems. Give one way that we have used equations in this class to express and extract information about a physical system and give a specific example. Discuss briefly whether your other classes have used equations in a similar way. Note: This is an essay question. Your answer will be judged not solely on its correctness, but for its depth, coherence, and clarity.

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5. Muscles consist of multiple elastic fibers connected together and to tendons in complex ways. To get a start on thinking about how these work, consider the behavior of two simple springs connected in series and parallel. For this problem, ignore the force of gravity on the springs.

At the right are shown two identical springs connected in a different way (in parallel). If these springs are well described by Hooke's Law, T = kΔL, for the relevant range of deformations, what would the effective spring constant for the combined pair be? (That is, putting a box around the pair and treating them as if they were a single spring.) Show your reasoning on the back and put your answer in the box below.

A. keff = k

B. keff = 2k

C. keff = k/2

D. Something else

The concentration of a chemical, A, is often written a "[A]". Since we are using square brackets to tell us to do a dimensional analysis, we will instead use the notation "cA" to represent the concentration of a chemical A. We express dimensionality to show how different measurements must be combined to create the quantity involved: in powers of M (measured by a scale), L (measured by a ruler), and T (measured by a clock).The dimensionality of a concentration is number/unit volume, [c] = 1/L3.

1.4 If the chemical reaction A + B forward

reverse AB has reaction rates kf for the forward reaction and kr for

the reverse reaction, then (in the absence of a catalyst) the rate of change of the concentration of the chemical AB is given by

dcAB dt

= −k f cAcB + krcAB

If this is so, are the dimensionalities of kf and kr the same?

1.5 What is the dimensionality of the rate constant kf ?

3. I saw a commercial for a health care product that said, "There are 10s of trillions of cells in the human body." This seems like a lot! A trillion is 1012. Estimate the number of cells in the human body to see if the statement is approximately correct or not. Be sure to clearly state your assumptions and how you came to the numbers you estimated, since grading on this problem will be mostly based on your reasoning, not on your answer.

3. Cells have a number of ways of communicating information within a cell and between cells. One such mechanism is diffusion. A signaling chemical is produced at one place and diffuses to another. But in some places, diffusion takes too long and other mechanisms had to evolve. In the figure at the right is shown a neuron (at the top) and its axon (the long cable) connecting to a second neuron (at the bottom). The small black blob in the middle of the top neuron is its cell body – about 10 μm in diameter. The coefficient of diffusion of a signaling protein in cytoplasm is about D ~ 10 μm2/s. Estimate about how long it would take a small blob of signaling chemical molecules from the upper cell's cell body to diffuse down the axon to the second cell. Be sure to clearly state your assumptions and how you came to the

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2. Ion channels that permit ions to pass between the interior of the cell and the extra-cellular medium are an important part of a living cell. There are multiple mechanisms that happen in an ion channel, but the simplest is that ions flow through the cell preferentially in one direction as a result of diffusion arising

from a difference of ion concentrations on the inside and outside of the cell.

As a first model of an ion channel, let's see how it would behave if the flux through the open channel were simply due to diffusion arising from a concentration difference. We'll model the open channel as a cylinder as shown in the figure at the left.

Fick's law for flow due to a concentration gradient is Jx = −D Δc Δx

, where D is the diffusion

constant and J has dimensions corresponding to number per unit area per unit time.

2.1 A typical ion channel radius is about 1 nm (nanometer), and the thickness of the cell membrane is about 7 nm. The concentration of Na+ ions inside a mammalian heart cell is about 0.6 ions/nm3, and in the fluid surrounding the cell is about 6.0 ions/nm3. The diffusion constant for ions is about 109 nm2/s (nanometers2/second). If the channel were just an open cylinder, how many ions per second would flow through the channel and in which direction?

2.2 What is responsible for the flow of the ions through the channel in this model?

A. The ions are attracted to the channel and since there are more on one side than the other there is a net flow from the more dense concentration side to the less.

B. The ions move randomly as a result of collisions with the fluid molecules and since there are more on one side than the other there is a net flow from the more dense concentration side to the less.

C. There are more ions in the high concentration region. Their collisions with each other drive the ions in all direction and since there are more on one side than the other there is a net flow from the more dense concentration side to the less.

D. The ions sense when the channel is open and explicitly move towards it.

2.3 Can this "diffusion constant, D" be the same constant as in the other Fick's law equation, x2 = 2DΔt ? Show whether they have the same dimensionality or not.

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3. Estimate how many helium balloons of the indicated size in the figure at the right would be needed to lift the kitten. You can take the density of air to be ~ 1 kg/m3 and the density of helium to be about 1/7 that of air when both are at STP. Do you need to worry about the weight of the balloons? Be sure to clearly state your assumptions and how you came to the numbers you estimated, since grading on this problem will be mostly based on your reasoning, not on your answer.

5. Two long molecules have unbalanced charges at the end so the field they produce looks like that of the indicated + and – charges of magnitude e at the ends of the molecules as shown in the figure at the right. The rest of the molecules’ charges essentially cancel and can be ignored. The molecules have a length d and are separated by a distance d. Express your answers in terms of the combination

F0 = kce

2

d 2 . Show your work.

5.1 If an ion of charge e is placed at the origin of the coordinate system (where the two axes cross), what is the magnitude of the electric force that the ion feels?

5.2 If an ion of charge +e is placed one the y axis right between the top two charges (at the location (0, d/2)), what is the magnitude and direction of the force that ion would feel from the molecules?

5.3 Now consider the forces that the two molecules exert on each other.

5.31 In what direction is the net force that the left molecule exerts on the right?

5.3.2 What can you say about the magnitude of the net force that the left molecule exerts on the right?

A. It is less than F0. B. It equals F0. C. It is between F0 and 2F0. D. It equals 2F0. E. It is greater than 2F0.

1. The figure at the right shows the graphs for the position of a small sphere dropped into a long cylinder filled with oil from a few inches above the oil's surface. A number of different times (labeled t0 to t3) are shown on the graphs.

1.1 At which time does the sphere enter the oil?

A. t0 B. T1 C. T2 D. T3 E. None of them.

1.2 At which time does the sphere appear to reach terminal velocity?

A. t0 B. T1 C. T2 D. T3 E. None of them.

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1.3 At time t3, what forces are there acting on the sphere?

A. The force of gravity and the buoyant force of the oil. B. None, since it is moving at a constant velocity. C. The force of its motion. D. The resistive force from the oil, gravity, and the buoyant force of the oil. E. The resistive and buoyant forces from the oil. F. Some other combination.

1.4 If scales for all of these axes were given, what would you need in order to find the net force acting on the object at each instant?

A. Only the mass of the object. B. Only the viscosity and density of the fluid. C. Both the viscosity of the fluid and the object's mass. D. No additional information is needed. It can be found from the graphs. E. Some additional information is needed.

3. Two long molecules have unbalanced charges at the end so the electric field they produce looks like that of only the indicated + and – charges of magnitude e at the ends of the molecules as shown in the figure at the right. The rest of the molecules’ charges essentially cancel and can be ignored. If a positive ion is added to the four charge system, it will add some electric potential energy. Rank how much electric potential energy will be added if the ion is placed at one of the five positions, A, B, C, D, and E. If any are 0, indicate that by including an "=0" in your ranking string.

4. At the right is shown a pipe with a uniform circular cross section carrying a fluid at a steady flow. The piece of the pipe we are considering has a length, L, and a cross sectional area, A. The fluid is flowing at a rate of Q cm3/s. The pressure at the beginning of the segment is p1, and at the end of the segment is p2. The fluid has a viscosity, μ.

4.1 Starting with the original pipe, if the pressures are maintained at the same values at the ends of the pipe segment but the pipe is replaced by a narrower one (smaller radius), what do you expect will happen to Q, the rate of flow through the pipe?

A. It will be greater than it was originally. B. It will be less than it was originally. C. It will stay the same as it was originally. D. There is not enough information given to decide.

4.2 If the pressures are maintained at the same values at the ends of the pipe segment but the pipe is replaced by a longer one, what do you expect will happen to Q, the rate of flow through the pipe?

A. It will be greater than it was originally. B. It will be less than it was originally. C. It will stay the same as it was originally. D. There is not enough information given to decide.

4.3 In the figure at the right are given four equations. One of them is the correct Hagen-Poiseuille equation. Which is it?

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II. Various components in a cell are moved through the cell by motor proteins such as kinesin. The kinesin attaches itself on one end to the packet it is moving (the "cargo" shown here as a big sphere) and the other end to a tubule (at the bottom of the picture). The kinesin has two "feet" at the bottom that detach, swing forward, and attach again alternately, "walking" the kinesin along the tubule. The cargo it's dragging is not moving in a vacuum, but is moving through a storm of wildly fluctuating fluid molecules (the water and other chemicals of which the fluid in the cell is made). This means that to drag the object through the fluid – even at a constant speed – the kinesin has to put in energy to counteract the resistive force of the fluid. It does so by burning ATP into ADP.

IV. Suppose the packet being dragged through the cell's fluid has a mass, m, and, when it is moving through the fluid with a speed, v, experiences a viscous force of magnitude f = bv. Let's make a simple model of how this works in order to get a sense of scale. If the kinesin is exerting a constant force, Fk, on the cargo, which of the following expresses the condition on the two forces in order that the cargo move at a constant velocity?

a) Fk > bv

b) Fk = bv

c) Fk < bv

B. How much work, ΔW, does the kinesin have to do in order to move the cargo a distance Δx along the tubule? Express your answer in terms of Δx and the symbols used in part A.

C. Using your result from B, create an expression for the rate at which work is done by the kinesin, ΔW/Δt.

D. A kinesin that is transporting a secretory vesicle uses approximately 80 ATP molecules/s. Each ATP provides a kinesin molecule with an energy of about 0.8 x 10-19 J. If the velocity of the kinesin is 800 nm/s, can you determine the force the kinesin is exerting, if you assume that all the ATP energy is used (100% efficiency)? If you can, find it. If not, explain why not.

III. In liquid water, water molecules are continually forming and breaking hydrogen bonds. In water, a hydrogen bond has a bond energy of about 0.2 eV. Two water molecules are shown forming a hydrogen bond (dashed line) in the figure at the right.

A. Is the formation of the hydrogen bond exothermic (releases energy, reducing the internal energy of the molecules) or endothermic (absorbs energy, increasing the internal energy of the molecule)? Explain briefly.

Let's see whether hydrogen bonds can account for the energy needed to boil water. Assume the model that when water boils, the energy that has to be put in is the energy needed to break the hydrogen bonds.

It takes about 2.3 x 106 J to boil 1 kg of water.

B.1 First, figure out how many water molecules there are in 1 kg of water. (Hint: The atomic mass of water is 18D and Avogadro's number is ~ 6 x 1023 molecules/mole

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B.2 Do you expect that there would be more hydrogen bonds than water molecules or fewer? Explain your reasoning.

B.3 Next, figure out in our model of boiling, how many hydrogen bonds are broken in order to boil 1 kg of water.

B.4 Does your calculation support our simple model or not? Why?

IV. The pH inside a biological cell is typically pretty close to 7. This means that the molar concentration of H+ ions is 10-7 mol/L. For a bacterial cell shown (staphylococcus aureus) estimate approximately how many H+ ions there are in the cell. Be sure to clearly state your assumptions and how you came to the numbers you estimated, since grading on this problem will be mostly based on your reasoning, not on your answer. (Figure source: USDA – emu.arsuda.gov)

Homework Problems

Scaling up Here are two related problems, one precise, one an estimation.

(a) A sculptor builds a model for a statue of a terrapin to replace Testudo.* She discovers that to cast her small scale model she needs 2 kg of bronze. When she is done, she finds that she can give it two coats of finishing polyurethane varnish using exactly one small can of varnish. The final statue is supposed to be 5 times as large as the model in each dimension. How much bronze will she need? How much varnish should she buy? (Hint: If this seems difficult, you might start by writing a simpler question that is simpler to work on before tackling this one.)

(b) The human brain has 1000 times the surface area of a mouse's brain. The human brain is convoluted, the mouse's not. How much of the increase in surface area is due to just size -- the human brain is bigger? How sensitive is your result to your estimations of the approximate dimensions of a human and mouse brain?

* Testudo is the statue of a Terrapin (the university mascot) in front of the main library on the University of Maryland campus.

A homeopathic drug

Flu season is approaching! Oscillococcinum is a homeopathic medicine that is advertised as relieving flu symptoms. It is produced by starting with duck liver and heart, and diluting them to “200C”. This means that they are diluted to 1 part per hundred, and then this is repeated 200 times. (Thus, after two dilutions, the ratio is 1 in 104; after three, the ratio is 1 in 106, and so on all the way to 200.) Estimate how many molecules of the original duck organs are in one Oscillococcinum pill.

Be sure to clearly state your assumptions and how you came to the numbers you estimated, since grading on this problem will be mostly based on your reasoning, not on your answer. We are not interested in whether you can “guess” a right answer, but in whether you can use your everyday experience to infer information about topics about which you may have little direct knowledge.

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Growing fingernails Chemotherapy is a treatment that cancer patients take that suppresses the growth of fast growing cells. While this often stops the cancer in its tracks, it has a number of common side effects since all fast growing cells are affected. Chemotherapy patients lose their hair, their tastebuds, and often their fingernails. A chemotherapy patient may well be interested in the answer to the question: "Once I stop my chemotherapy, how long will it be before my fingernails grow back? A few days? A few weeks? Months? A year?" (It's hard to scratch an itch without fingernails!)

From your personal experience, estimate how long it will take a chemotherapy patient to grow back their fingernails. Be sure to explain what assumptions you are making and how you are quantifying your personal experience.

Dimensions of pressure The change in pressure, Δp, as one descends under the surface of water depends on the density of water, ρ, the gravitational field, g, and the depth, d, beneath the surface. An equation for Δp will have all these three quantities in it (and no other dimensioned quantities). The various quantities have the following dimensionalities:

• [Δp] = M/LT2 • [ρ] = M/L3 • [g] = L/T2 • [d] = L

A. We expect that the change in pressure, Δp, will be proportional to some power of d – call it dn where n might be positive or negative. Only reasoning using dimensionalities, can you determine n?

a. No you can't. b. I'm sure you can, but I don't know how to do it. c. Yes you can.

B. If you chose "Yes you can" to the first part, give the value of n. If you chose another answer, explain why you chose that answer.

The tortoise and hare A speedy tortoise can run with a speed of vT and a hare can run with a speed vH. The hare waits to rest for a time Δt0 after the race begins and the tortoise starts and then runs as fast as he can, but the tortoise still wins by a shell (length of shell = s). The length of the racetrack is a distance, d. The race is considered over when the tortoise crosses the finish line.

a. Using the symbols of the problem, write an equation for the amount of time that the tortoise ran, ΔtT.

b. Using the symbols of the problem, write an equation for the amount of time that the hare ran, ΔtH.

c. If you only know the speeds of the tortoise and the hare, the rest time and the length of the shell, can you find the length of the racetrack and the times that the animals each ran?

Finding a small tumor Some cancers can be detected by a simple blood test. The cancer cells secrete a chemical that circulates in the blood. A measure of the amount of that chemical tells the amount of cancer cells in the body.

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One example of this is prostate cancer. The prostate is a gland that often becomes cancerous in men as they age. Cancerous prostate cells secrete a prostate specific antigen or PSA. A blood test for PSA is recommended for men over the age of 50 and is often used as a first warning of possible cancer.

One way of dealing with prostate cancer is to surgically remove the gland (prostatectomy). After that has been done, there should be no cancer cells left and therefor no PSA found in the blood. Since it's hard to get all the cancer cells and some may escape, after a prostatectomy, regular tests for PSA are done to detect possible presence and growth of cancer cells. Another way to look for such metastatic cancer is with an MRI (Magnetic Resonance Imaging) or CT (Computerized Tomography) scan. In this problem, we'll estimate which method is more sensitive.

A spherical tumor 3 cm in diameter consisting of prostate cells may result in a PSA blood level of about 20 nanograms/milliliter of blood. The test can only detect the presence of PSA in the blood if it is greater than about 0.1 ng/ml. An MRI can detect a cancerous tumor down to about a diameter of 1 mm.

Which is a more sensitive test for the return of prostate cancer, the blood test or an MRI? To decide this, estimate how big a tumor would produce a PSA level just less than 0.1 ng/ml. Would a tumor that produced this (undetectable) amount of PSA be visible on an MRI?

Acceleration at the top Two physics students were discussing a problem in which the class was asked to find the acceleration of a cart rolling up and down an incline at the instant the cart was at the very top of its path. Malia says, “At the top the velocity has to be zero so the acceleration has to be zero too.” Sasha disagrees saying, “No, the velocity is changing at the top so the acceleration can’t be zero.” Who do you agree with? Discuss the reason for your answer and explain why the one who is wrong might legitimately be confused about which answer is correct. Note: This is an essay question. Your answer will be judged not solely on its correctness, but for its depth, coherence, and clarity.

How fast do you need to go? A. I need to drive from A to C via B at total distance of ΔL. The distance from A to B is ΔL1, from B to C is ΔL2. I have time Δt to get from A to C, so I figure I can make it if I average = ΔL/Δt for the entire trip. Going from A to B the traffic is heavy and I can only average a speed v1. How fast do I have to go from B to C in order to average for the whole trip? Write an equation expressing v2 (the speed I need to average going from B to C) in terms of the other symbols given: ΔL, ΔL1,ΔL2, and Δt.

B. I need to drive from A to C via B. The distance from A to B is 25 miles, from B to C is 75 miles. I have 2 hours to get from A to C, so I figure I can make it if I average 50 mi/h for the entire trip. Going from A to B the traffic is heavy and I can only average 25 mi/h. How fast to I have to go from B to C in order to average 50 mi/h for the whole trip?

1. 58 mi/h 2. 60 mi/h 3. 75 mi/h 4. 100 mi/h 5. Something else 6. You can’t get it from the information given.

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Slime mold spores

Slime mold cells feast on bacteria and yeast and essentially any food that comes their way. However, if they are out of food, they signal to each other and move together with about 100,000 of their best friends to form a spore. A typical spore is shown on the right. Unfortunately the image does not have a scale bar so we are not sure how large it is. Someone (Juan Lasheras, UCSD) did, however, take a picture of a single cell as shown on the right, with a scale bar of 5 microns. Estimate how heavy a spore is (in Newtons). (Note that the spores are just large enough to survive as a group yet light enough and high enough above the surface to be carried away by the wind!) Be sure to clearly state your assumptions and how you came to the numbers you estimated, since grading on this problem will be mostly based on your reasoning, not on your answer.

Source: J. Lasheras, UCSD

Pushing a block on a block (symbolic)

A heavy block, labeled “A”, is sitting on a table. On top of that block is a lighter block, labeled “B” as shown in the figure at the right. For the first parts of this problem you are asked to identify the direction of forces in this system under various circumstances. In this problem, we will be looking at the relationships between the various forces in the problem under various circumstances. In order to simplify the equations we write, we will not use our full "who is acting on whom" force notation, but will use the following simplifications. (Note that we have taken for granted that you understood and could use Newton's 3rd law.)

• NF→A = F (force of the finger pushing block A) • NA→B = NB→A = N (the normal forces acting between the blocks) • NT→A = NT (the normal force of the table acting on block A) • fT→A = fT (the friction force between block A and the table) • fA→B = fB→A = f (the friction force between the two blocks) • WE→A = WA (weight of block A) • WE→B = WB (weight of block B)

We then have the seven symbols representing all the forces in the problem: F, NT, N, fT, f, WA, and WB.

(A) If the finger is pushing but not hard enough, neither block will move. By Newton's 2nd law, for any object that is not accelerating, the forces in each direction must balance.

1. Write the equations for the balance of the forces in the horizontal and vertical directions for block A and for block B (four equations).

2. Three of the quantities are in principle straightforward to measure (using, say, a spring scale): F, WA, and WB. The other four, fT, f, NT, and N, are "invisible" -- that is, a little bit harder to

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measure directly. (Though you might be able to think of a way!) If you knew the three easily measurable quantities, could you find the other four (invisible) ones? If so, write equations to express each invisible one in terms of the three measurable ones. If not, explain what else you would need.

(B) If the finger is pushing hard enough, the two blocks will start to speed up. Assume they speed up together without slipping. By Newton's 2nd law, for any object that is accelerating, ma in each direction must equal the net forces in that direction.

1. Write the equations for ma in the horizontal and vertical directions for block A and for block B (four equations).

2. In addition to the three quantities that are in principle straightforward to measure (F, WA, and WB), the accelerations aA and aBcan be easily measured (say using a video capture program). If you knew the accelerations and the three easily measurable quantities, could you find the other four (invisible) ones? If so, write equations to express each invisible one in terms of the three known quantities. If not, explain what else you would need.

3. Compared to part (A), we have new quantities, but the same number of equations. How is this possible? What changes between the two situations?

(C) Once the blocks have sped up, the finger still has to push on it to keep it going at a constant velocity, even though N2 tells us that at a constant velocity, all forces must balance. Assume that both blocks move with the same speed and don't slide.

1. Write the equations that express the balance of forces for each block and each direction.

2. Suppose we assume that the finger is pushing harder on the blocks in part (C) than it did in part (A) but not as hard as it was in part (B). Which of the other 6 forces have to change compared to what they were in part (B)? If they do change, do they each get bigger? small? go to zero?

Equations of counting A. Stoichiometry

One of the most important reactions in biology is photosynthesis. In this complex chemical process, plants combine molecules of carbon dioxide and water to create glucose and oxygen. The reaction looks something like this:

CO2 + H2O → C6H12O6 + O2

Of course the number of atoms don't balance on the two sides. You might be able to intuit your way to the correct counting (stoichiometry), but let's do it systematically by setting up equations.

1. Write the reaction as

a CO2 + b H2O → c C6H12O6 + d O2

where a, b, c, and d are meant to be integers adjusted so that the number of each kind of atoms balance on each side. For this reaction, we have three kinds of atoms that must balance: carbon (C), hydrogen (H), and oxygen (O). Write three equations for a, b, c, and d that, if satisfied, will guarantee that the number of atoms of each kind are the same on each side of the reaction.

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2. Find the smallest set of integers that will satisfy these equations and write the correct stoichiometric equation for photosynthesis.

3. We only had three equations but we had four unknowns. That usually produces an undetermined situation in math. How did we find a unique solution?

B. Dimensional analysis

When we do dimensional analysis, we do something analogous to stoichiometry, but with multiplying instead of adding. Consider the diffusion constant that appears in Fick's first law:

In this expression, J represents a flow of particles: number of particles per unit area per second, n represents a concentration of particles: number of particles per unit volume; and x represents a distance. We can assume that they have the following dimensionalities:

• [J] = 1/L2T • [n] = 1/L3 • [x] = L

1. From this, determine the dimensionality of D.

2. Einstein discovered a relation that expresses how D depends on the parameters of the system: the size of the particle diffusing (R), the viscosity of the fluid it is diffusing in (μ),and the thermal energy parameter (kBT). These have the dimensionalities

• [kBT] = ML2/T2 • [μ] = M/LT • [R] = L

Assume that we can express D as a product of these three quantities to some power, like this:

D = (kBT)a (μ)b (R)c

Write equations for a, b, and c that will guarantee that D will have the correct dimensionality for M, L, and T.

3. Solve these equations and write an expression how D depends on the three parameters. (The correct equation has a factor of 1/6π that cannot be found from dimensional analysis.)

In this case, there were exactly the right number of parameters to give the same number of equations as there were unknowns. That will not always be the case, but sometimes this method still can be made to work by adding physical knowledge about the dependences.

Propelling a paramecium – estimations a. We are going to solve for the motion of the paramecium using a spreadsheet (like Excel or

GoogleDocs) and a stepping rule. To do that we need to choose a time step dt. In any problem, we can get an idea of what the appropriate time scale is by creating a time out of the parameters. By considering the dimensionality of the parameters β, m, and R [dimensionality means, M,L,T and their combination -- see the page Complex Dimensions and Dimensional Analysis] can you combine the

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three of these parameter as products or ratios so that the resulting combination has the dimensions of time? This combination gives a natural time scale for the problem.

b. Let's evaluate the natural time scale for the paramecium's motion. The value of β for water at room temperature is about 0.017 kg/m-s. But we also need the mass of the paramecium. Although a paramecium is stretched out a bit, let's take an oversimplified model treating it like a sphere. An average radius for a paramecium is about R = 1/4 mm. Estimate the mass, m, of such a paramecium. Discuss what density you choose and why. Combine these numbers to find the natural time scale for the motion of the paramecium.

But it's just a leetle bit different We know that within the limits of measurement, the magnitudes of the negative charge on the electron and the positive charge on the proton are equal. Suppose, however, that the magnitude of the charge on the proton was bigger than that on the electron by 1 part in a million. Estimate with what force two copper pennies, placed one meter apart, would then repel each other. You may find some of the following parameters useful (or not). Here are some potentially useful parameters:

kC (Coulomb's constant) ~9 ✕ 109 N-m2/C2 e (charge on proton) ~1.6 ✕ 10−19 C

me (mass of electron) ~9 ✕ 10−31 kg mp (mass of proton) ~1.7 ✕ 10−27 kg

mCu (mass of a copper atom) ~ 10−25 kg Number of electrons in a copper atom = 29

Since pennies do NOT repel each other with any detectable force, what can you conclude about the charges on a proton and on an electron?

Colliding carts

Two carts are riding on an air track as shown in the figure at the right. At clock time t = 0 cart B is at the origin traveling in the +x direction with a velocity speed vB0. At that time, cart A is at the position shown and is at rest. Cart B has twice the mass of cart A. The carts "bump" each other, but don't stick.

In the graphs below are shown a number of possible plots for the various physical parameters associated with the two carts. Each graph has two curves, one for each cart and labeled with the cart's letter. For each property (a)-(e) select the letter of the graph that could be a plot of the property.

a. The forces exerted by the carts

b. The position of the carts

c. The velocity of the carts

d. The acceleration of the carts

e. The momentum of the carts.

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Random or not, here I come. In his physics lab, Radagast has observed the motion of an E. coli bacterium using a video camera. His log-log plot of the square deviation of his chosen bacterium as a function of time is shown in the figure below. (The graph is presented in two equivalent forms: one for those who have used Excel – on the left – instead of log-log graph paper – on the right.) The bacterium seems to have two distinct behaviors: for times shorter than 1 second (marked A) and for times longer than 10 seconds (marked B). What do you think might be an appropriate hypothesis for what might be causing the two different behaviors?

1. In region A:

A. The bacterium is moving purposefully in response to some chemical gradient. B. The bacterium is moving at random in response to the thermal motion of its environment. C. The bacterium is constrained in some way. D. The bacterium is using its flagella (which work like propellers) to move at a constant velocity E. The bacterium is accelerating in response to a force in a fixed direction. F. None of these behaviors are consistent with that part of the graph.

2. In region B:

A. The bacterium is moving purposefully in response to some chemical gradient. B. The bacterium is moving at random in response to the thermal motion of its environment. C. The bacterium is constrained in some way. D. The bacterium is using its flagella (which work like propellers) to move at a constant velocity. E. The bacterium is accelerating in response to a force in a fixed direction. F. None of these behaviors are consistent with that part of the graph.

Diffusion in time and space A. A common small-molecular weight (and therefore fast diffusing -- for an organic molecule) ingredient in perfumes is vanillin, the primary component of vanilla bean extract (molecular weight = 152). The D for vanillin in air is 0.114 cm2/s. If I open a bottle of vanilla on the other side of the room 3 meters away, and the air is still so there is no convection, about how long would I have to wait before I could expect to smell the vanilla?

B. The diffusion constant for glucose in water is 0.70 x 10-5 cm2/s. About how much time would it take to get significant diffusion across the following structures? Assume that glucose diffuses through cells and membranes as fast as it does through water.

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1. Across a cell membrane 8 nm thick? 2. From the center of a eukaryotic cell 6μm in radius to the cell membrane? 3. Across the wall of the human heart 2.5 cm in thickness? 4. From the cells in the center of the cactus to the photosynthetic cells on the surface of a barrel

cactus if the cactus has a radius of 12 cm?

Molarity and number density Density is a very important concept in physics. It tells how much mass is packed into a unit volume. But in chemistry, since atoms interact individually and not according to mass, the "density" that matters is the count -- the number of atoms of a particular chemical per unit volume -- a number density. In this problem we'll explore how a number density gives us information on how close molecules are to each other -- how much space each individual molecule gets. This gives us a good insight to how to think differently about gases and liquids on the molecular level.

A. The 'number density" in chemistry is molarity: the number of moles of a particular chemical in a given volume. The particular choice made in chemistry is typically the number of (gram) moles of a particular chemical in a liter. Physicists tend to be more interested in molecules and in metric units. Given that a mole of a chemical means Avogadro's number of molecules (6 x 1023) and that a liter means 1000 cm3 (since one thousandth of a liter -- one milliliter = 1 cm3), it's reasonably straightforward to convert from moles/ml to molecules/cm3. Let's write that m = molarity in units of moles/ml and n = number density = N/V, the number of molecules divided by the volume they are in, in units of molecules/m3. Since m and n must be proportional, we expect there is an equation

n = αm

where α is some constant. Find the numerical value of α and its units (keeping "moles" and "molecules" as units).

The bigger n is, the closer together are the molecules. It's useful to also consider the reciprocal of this -- the volume divided by the number of molecules. This gives the amount of volume each molecule occupies by itself, on the average. ("By itself" meaning with no other molecules of the same type. Of course there may be other molecules in this volume, but on the average it gives a sense of how far apart each molecule is from others of its kind.) This quantity, s = 1/n = V/N, is called specific volume, or we might call it separateness. Larger s means that each molecule occupies more volume so they are more separated, while smaller s means each molecule occupies less volume so they are closer (less separated). Note that here "occupies" means "moves through by itself", not the actual volume the molecule takes up.

B. In chemistry we learn that at STP (standard temperature and pressure), 22.4 liters of gas contains one mole of molecules. Given that air has a density of 1.225 kg/m3, and it is about 20% oxygen (O2) and 80% nitrogen (N2), find the molarity and specific volume for molecular oxygen and nitrogen in the air. An oxygen molecule has a molecular weight 0f 32 D and nitrogen has a molecular weight of 28 D.

C. Given that the molecular weight of water (H2O) is equal to 18 D, and that the density of water is 1 gram/ml, find the molarity and specific volume of water.

D. Given that the molecular diameter of a one molecule of molecular oxygen, molecular nitrogen, and water are all about 0.12 nm, what fraction of their individual volumes does each of them fill?

E. If we want a sense of how far apart molecules are, it's more convenient to have a distance, not a volume. To get a distance (a length) from a volume (a length cubed), we need to take a cube root. Is the distance,

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d = s1/3

where s = 1/αm a reasonable estimate for the average separation between molecules of molarity m? Discuss why you do or do not think so.

F. Find the values of d for oxygen and nitrogen molecules in air at STP and for liquid water.

Kinetic theory and pressure

A box of gas contains a mixture of two different gases, shown in the figure as blue molecules and red molecules. The temperature and pressure in the box are measured at STP (T = 300 K and p = 1 atmosphere). The blue molecules have 4 times the mass of the red molecules, but there are equal numbers of each kind in the box. The simulation has been running long enough at this time so that the all the measurements have stabilized. (Gravity is turned off in the simulation.)

1. Which molecules contribute more to the pressure measured by a meter in the wall of the box?

2. How do the temperatures of the molecules compare?

3. Which molecules are moving faster?

An additional amount of red molecules are introduced until the box contains 4 times as many of the red molecules as there are of the blue as shown in the figure at the right.

4. Now how do the average speeds of the molecules compare?

5. How do the temperatures of the molecules compare?

6. On the average, how do the contributions of the molecules to the force felt on the left wall compare?

What if the Greenland ice sheet melts? An important question to consider when thinking about global warming is, "If the ice sheets near the poles melt, how much will the sea level rise?" This seems like a difficult question, given the odd shapes of both the ice sheets and the oceans. But there are some accurate approximations that allow the answer to be estimated fairly accurately with reasonably simple calculations.

The critical idea is that both the thickness of the ice sheets and the amounts of the sea-level rise are extremely small compared to the radius of the earth. The radius of the earth is about 2/π x 107 m -- more than 6000 kilometers. The ice sheet thicknesses we will be concerned with are single digit miles and the sea level rises will be in dozens of feet. As a result, in thinking about them, we can essentially ignore the curvature of the earth. We can imagine peeling the map of the earth off a globe and flattening it out (by

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making cuts, not by stretching it, so that we preserve the area). Then, both the ice and the sea level rise can be treated as right (not tilted) cylinders (though with funny shaped bases and tops). Since we know that the volume of a right cylinder is the area of the base times the height, we can easily estimate all the volumes we need. A schematic picture of this approximation (with the height of the ice mass greatly exaggerated -- you couldn't see it if I didn't) is shown below. The error in these approximations is on the order of the height of the cylinder considered divided by the radius of the earth; a very small number. (For your calculation, estimate the real volume of the ice. Don't treat it as a circular cylinder!)

a. Assume that (after flattening the surface of the earth in our imaginations) the ice sheet to be considered covers an area, A,and has a thickness, d. The oceans currently cover about 75% of the earth's surface, and note that the surface area of a sphere is 4πr2where r is the radius of the sphere.

Generate an equation that will allow you to calculate h, the height the ocean levels will rise due to the melting of a an ice sheet in terms of A, d, and r

b. As shown in the combined satellite photo on the right (from Google Earth), Greenland is covered by a sheet of ice. This ice has been measured to have a mean thickness of about 2 km. Recent observations indicate that this ice sheet is beginning to retreat -- that it is melting at an accelerating rate.

Using the depth of the ice sheet and the scale given on the picture, estimate how much sea level rise would be produced by the melting of the entire ice sheet lying on Greenland. (Note: You can easily look this information up on the web. But the goal of this problem is in part to develop your estimation skills for the purpose of building your ability to decide for yourself whether any given piece of information you find on the web is reasonable or bogus.)

c. Find a website that allows you to see what the coast of Florida would look like if the sea level rose the amount you calculated. Include a printout of this picture as part of your solution and give a brief reason why you trust this site.

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How many ions?

When the number of molecules, atoms, or ions of a particular type (here we consider H+ ions) is in the hundreds of millions, we can confidently speak of concentrations and analyze an average behavior. However, when the number is very small, random fluctuations in the actual number present become important.

As a rule of thumb we need a different type of description (stochastic equations!) when describing 10 or fewer atoms, molecules, or ions. Consider two compartments of comparable size but very different function inside eukaryotic cells: mitochondria and lysosomes (#9 and #12 in the schematic at right). These have very different pH from the rest of the cell. Mitochondria have a pH of ~8. This means that there are about 10-8 moles/liter of H+ ions. Lysosomes on the other hand have a pH of ~5.

Figure source: Wikimedia Commons

Estimate whether we need to consider “stochastic equations” in either compartment. Be sure to clearly state your assumptions and how you came to the numbers you estimated, since grading on this problem will be mostly based on your reasoning, not on your answer.

Springy cart of a track

A cart with a spring on the end of it is moving to the right (in the positive x direction using the coordinate system shown) on a frictionless air track as shown in the figure at the right. It keeps going until it hits a wall, the spring compresses, and the cart bounces off the wall and moves in the opposite direction. Choose all the graphs that could represent each of the following quantities as a function of time. (In each graph, the axes cross at the origin of the two coordinates.)

6.1 Position of the cart 6.4 Kinetic energy of the cart 6.2 Velocity of the cart 6.5 Potential energy of the spring 6.3 Momentum of the cart 6.6 Total mechanical energy of the cart (including the spring)

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Sliding down a ramp

A small block of mass m is place on top of a hill as shown in the figure at the right, a height h above a horizontal track. The hill is frictionless, but when the block reaches the horizontal track, there is a coefficient of friction, μ. The block slides along the track for a distance L and comes to a stop. For this motion air resistance can be ignored. If we take the origin of the coordinate system to be at A, with x being the horizontal coordinate with positive to the right, and y being the vertical coordinate with positive up, the shape of the hill (for d1 > x > 0) is given by the function

1. The task is to find an equation expressing the distance L in terms of the parameters of the problem: m, g, μ, d1, d2, h, and θ. Which of the following statements is true?

A. There is not enough information to find L. Additional information is required. B. There is enough information to find L.

2. If you chose answer A for problem 1, explain what information would be needed. If you chose answer B, write an equation that would allow you to find L in terms of the parameters of the problem.

Energy in photosynthesis – a toy model Photosynthesis is the process by which plants effectively capture energy from the sun and store it for later use by organisms. It's the fundamental process that enables the entire food chain. Actual photosynthesis is a highly complex process involving many steps and many different chemicals. But the idea of what is happening to the overall energy balance can be seen even in simpler reactions.

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Let's consider a toy model that shows how an input of energy (for example from the sun) could be used to combine water and carbon dioxide to create an organic molecule (formaldehyde) and oxygen and effectively "store" that energy for use later. The figure below shows a the simplest possible such process. The potential energy (binding deficit) of each bond is shown on the bonds.

A. First check the stoichiometry. Are the number of each kind of atoms balanced?

B. After you have balanced your equation, calculate the amount of bond energy deficit on each side of the equation and identify which side needs to have energy added to it to make the equation balance energetically and how much. What does it mean to say "you have to add energy to one side of the equation to make the energy balance?"

C. A particular direction of the reaction is labeled as exothermic. Is it correct? Explain how you know using some schematic visual representation of the energies (potential energy graphs or energy bars).

D. Discuss where the energy is "stored". This is not a question with a clear answer, since the energy relations in the reaction are a result of all the different bonds in all the different molecules. But discuss the two questions, "When the reaction goes in the exothermic direction and energy becomes available for use, where does the energy come from?" and "Where is it useful to think of the energy as being stored and why?" If these two answers are different, reconcile why it is OK for them to be different.

A superelastic collision Two gliders are set up to collide on an air track as shown in the figure below. The glider on the right is at rest and has a spring connected to it. Before the collision, as shown in the figure at the left, the smaller glider (mass m) is on the left approaching the larger glider (mass M). The spring on the larger glider is compressed and stores a spring potential energy of ΔE. When the two carts collide, the spring is released, giving all its potential energy up to the kinetic energy of the two carts. (The mass of the spring and latch is included in M; the friction internal to the spring and between the carts and the air track may be ignored.)

a. Is the momentum of the two carts conserved in this collision? If it is, write the equation that the variables of this collision satisfy as a result using the symbols given in the description of the problem. If it is not, explain why not.

b. Is mechanical energy conserved in this collision? If it is, write the equation that the variables of this collision satisfy as a result using the symbols given in the description of the problem. If it is not, explain why not.

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c. Collisions like this in which there is more KE in the final state than in the initial state is called a superelastic collision. Can you think of a common example in biology where more energy comes out than was put in? Discuss.

Potential energy of charge clusters Although the electric potential energy between a pair of charge has a quite simply form

UqQ = kCqQ/r

we rarely have a situation in which we only have two charges. Often we want to find the energy that results from adding a charge to a system of existing charges. In this problem we will consider two of the simplest systems that are more complex than a simple pair of charges: (1) a test charge probing the field of a dipole, and (2) a test charge probing the field of a quadrupole. The first is important in understanding how charges behave within a polar liquid (like water). The second is important in understanding how charges move when an action potential travels down an axon.

To do this, we'll use the simulation Charges and Fields from the PhET group at the University of Colorado. (I recommend clicking on the triple-bar menu icon in the lower corner, choosing "Options" and clicking of the "Projector mode" box so the screen is displayed on a light rather than a dark background.)

In the controls box at the upper right unclick the "Electric field" box and click the "Grid" box.

1. Put a pair of charges at the center of the screen separated by two large grid boxes. Stretch the measuring tape for 400 cm along the grid line above the charges and centered on them as shown below. The potential energy meter measures the extra energy (in electron Volts -- eV) that would be added to the system if a single proton was placed at the crosshairs.

1.1 Pick a coordinate system along the measuring tape. Create a table and measure the PE at various points along the measuring tape and graph the extra PE added as a function of position along that line. (Either use graph paper or a spreadsheet. Include the graph in your HW.)

1.2. Explain physically why the graph looks like it does.

1.3. Now put the tape measure all the way on the left and put the dipole on the tape measure line as shown in the figure at the right. Measure the PE of an added charge along the tape measure's line beginning 3 boxes to the right of the dipole and extending as far out to the right as you can go. Make a log-log plot of your results. If the field far from the dipole went like 1/r, the slope of your log-log plot would be -1. What is the slope you find? (Include your graph in your report.)

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1.4. Can you explain physically why the slope is NOT -1?

1.5 If the charges each have magnitude e and the separation of the dipole charges is a distance d, write an equation for the PE of the extra proton (charge +e) a distance x along the tape measure in 1.3 taking the origin as the center of the dipole.

2. Now create a quadrupole by taking two dipoles and putting them back to back as shown.

2.1 Measure the PE of the added proton along the line of the measuring tape and include a table of your data and a graph of the result in your report.

2.2 If the charges each have magnitude e and the separation of the quadrupole + and - charges are a distance d, write an equation for the PE of the extra proton (charge +e) a distance x along the tape measure in 1.3 taking the origin as the place where the vertical line through the center of the dipole crosses the measuring tape.

2.3 Can you explain physically why it looks the way it does?

Cells as Laplace bubbles 1. A bubble in a fluid system is described by the following variables:

• F = Force • A = Area • 𝛾 = Surface tension • r = Radius • p = Pressure

Which of the following relations are dimensionally correct?

2. In some cases it is useful to model a human cell as a semi-elastic membrane surrounding a colloidal fluid and in the body, under constant external pressure. Using this model, and the Laplace Pressure equation, if both the internal pressure of the cell and the radius of the cell are increasing, what is happening to the surface tension? Select all that could possibly occur.

a. Surface tension is increasing

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b. Surface tension decreases c. Surface tension remains constant

3. If, using the same model described in Problem 2, the cell’s radius is increasing and the surface tension is remaining constant, what is happening to the internal pressure? Select all that could occur.

a. Internal pressure is increasing b. Internal pressure is decreasing c. Internal pressure is remaining constant

4. If, using the same model described in Problem 2, the internal pressure of the cell in increasing and the radius of the cell is decreasing, what is happening to the surface tension? Select all that could occur.

a. Surface tension is increasing b. Surface tension decreases c. Surface tension remains constant