Physics 10424 Exam 1 February 10, 2012

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Physics 10424 Exam 1 Formulas

g = 9.8m/s2

Vectors : | �B| =�B2

x +B2y +B2

z i = x; j = y; k = z �vA,C = �vA,B + �vB,C

Kinematics : Constant ax : x = x0 + v0,xt+12axt

2 vx = v0,x + axt v2x = v20,x + 2ax(x− x0)

3D : �v =d�rdt = vxi+ vy j + vzk =

dxdt i+

dydt j +

dzdt k �a =

d�vdt = axi+ . . . = dvx

dt i+ . . . = d2xdt2

i+ . . .

at2 + bt+ c = 0 → t =− b±

√b2 − 4ac

2a; Newton :

� �F = �Ftot = m�a

Work : Won object =� �Fon object · d�r ∆U = −W ∆K = W ∆Ugravity = mg∆h

Power = �F · �v = �τ · �ω = W/∆t = dW/dt Fx, spring = −kx Uspring =12kx

2 Fcentrip =mv2r

Conservation of Energy : Kf + Uf = Ki + Ui +Winto system Klin =12mv2 F = −dU

dx

|�Fkinetic friction| = µkFnormal |�Fstatic friction| ≤ µsFnormal �p = m�v �J = ∆�p = �pf − �pi

Systems : Mtotxc.m. =�

i mixi �ptot = Mtot�vc.m. �vc.m. =d�rc.m.dt Ugrav = Mtotgyc.m.

Rotation : α = const. ⇒ ω = ω0 + αt ω2 = ω20 + 2α(θ − θ0) θ = θ0 + ω0t+

12αt

2

1 rev. = 2π rad�

τ = Iα vt = rω at = rα ar = rω2 I =�

i mir2i

Krot =12Iω

2 �L = �r × �p = I�ω τ = �r × �F = |�r||�F | sinφ = r⊥F Iparallel = Ic.m. +Mh2

�Fnet =d�Pc.m.dt �τnet =

d�Lc.m.dt Elastic : vi + vf = Vi + Vf f = ω/2π f = 1/T

Fgrav =GM1M2

r2 Ugrav = −GM1M2r (U = 0 at r = ∞) G = 6.67× 10−11Nm2/kg2

Fluid Statics : p2 = p1 + ρg∆h Pascal : p1 = p2 ⇒ F1A1

=F2A2

Archimedes : Fb = ρWgVdispl

Sphere : Area = 4πr2 Volume =43πr

3 Density : dm = ρdV

Fluid Dynamics : Continuity : A1v1 = A2v2 Bernoulli : p1 + ρgy1 +12ρv

21 = p2 + ρgy2 +

12ρv

22

Some Constants:

Mass of the Earth 5.98× 1024 kg Radius of Earth 6.37× 106 m

Mass of the Sun 1.99× 1030 kg Radius of Earth’s Orbit 1.5× 1011 m

Density of Water 1000 kg/m3

INSTRUCTIONS: Write your NAME on the front of the blue exam booklet. The exam is closed book, and you may have only pens/pencils and a calculator (no stored equations or programs and no graphing). Show all of your work in the blue book. For problems II-V, an answer alone is worth very little credit, even if it is correct – so show how you get it. Do not assume ANYTHING is obvious – if it’s not clear, please ask. It’s hard to write these questions, and we often forget things. Suggestions: Draw a diagram when possible, circle or box your final answers, and cross out parts which you do not want us to consider.

Physics 10424 Exam 1 February 10, 2012

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I. Multiple Choice Questions (4 points each) Please write the letter corresponding to your answer for each question in the grid stamped on the first inside page of your blue book. No partial credit is given for these questions.

1. The gravitational force of a strange extra-solar planet of mass ME on a satellite of mass m is ! = !!!!!!

. If launched from the planet’s surface, radius RE, what escape velocity ve would a rocket need to escape the gravitational field of the planet?

A. !! =  !!!!!!

! B. !! =  !!!!!! C. !! =  

!!!

!!!

D. !! =  !!!

!! E. !! =   !!!

2. In a cosmic game of “corn hole”, a satellite is tossed with velocity v an infinite distance away from a solid planet of uniform density ρ and radius R. The planet has a small hole drilled completely through the planet. The satellite, amazingly, flies straight through the planet without hitting the sides of the hole. Where during its trajectory does it have the maximum speed? Assume the planet is the only gravitational influence in the problem.

A. At the start. B. Upon first reaching the surface, at radius R. C. At the center of the planet. D. Upon reaching the surface of the planet on the other side of the center, at radius R. E. Far from the planet moving away from it.

3. A supertanker filled with oil has a total mass of 6.1 × 108 kg. If the dimensions of the ship are those of a rectangular box 300 meters long, 80 meters wide, and 40 meters high, determine how far the bottom of the ship is below sea level. (ρsea = 1.020 × 103 kg/m3.)

A. 10m B. 15m C. 20m D. 25m E. 30m

4. Water flows at speed v in a pipe of radius r. Neglecting viscosity, at what speed does the water flow through a constriction in which the radius of the pipe is r/3?

A. v/9 B. v/3 C. v D. 3v E. 9v

5. Fluid flows out of a large tank through the strangely-shaped pipe, as shown. At which point in the pipe is the pressure the greatest?

(continued)

• A • B

• D • C Flow

Physics 10424 Exam 1 February 10, 2012

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Problems (20 points each) II. Lagrange discovered that there are a certain number of points around the orbit of the earth where objects can orbit the sun at the same frequency as the earth, and thereby stay a stable, fixed distance from the earth and the sun. These places, called “Lagrange points” are important for observational satellites like the WMAP mission looking at the cosmic microwave background, which sits at Lagrange point 2, or L2. L2 sits on a line joining the centers of the earth and sun, outside the earth’s orbit. Take as constants R, the radius of the earth’s (circular) orbit, ME, the mass of the earth, and MS, the mass of the sun.

a) Find the speed of the earth in its orbit around the sun. b) Using the speed from part (a), find the period of the earth’s orbit around the sun c) Now, assume L2 is a distance r beyond the earth’s center. If a satellite is to stay at L2, it must

orbit the sun with the same period as the earth. Use the expression from part (b) to find the orbital velocity of the satellite around the sun in terms of r and other given constants.

d) Using this speed, write, but do not solve, an equation relating the Gravitational pull on the satellite at L2 to the force necessary to keep it in a circular orbit. (Simplify, but do not solve. You will end up with an equation cubic in r – best solved numerically or with some approximations…)

e) In one or two sentences, answer the following question: how can the stable orbit at L2 be possible?

III. Now, let’s say we would like to send a new observatory to L2. (The Webb Space Telescope is supposed to go there once it is launched, for example.) Typically, one would first put the observatory, of mass 5000 kg, into a low circular orbit around the earth. Take the orbit to be 100km above the earth, (RE = 6.37×106 m). From this point, we want to fire a boost rocket, changing the velocity of the probe so that the apogee of its new orbit will be at L2, a distance 1.496×109 m from the Earth’s center.

a) Find the speed of the probe in its original low-earth orbit, assuming ME = 6.0×1024 kg. b) Find the gravitational potential energy of the probe in its original orbit. c) Find the angular momentum of the probe in its original orbit, before the booster fires. d) What must be the velocity of the probe immediately after the booster shuts off so that it reaches

L2 at the apogee of the new elliptical orbit? (Hint: what quantities are conserved in the orbit?) e) Bonus: The boost rocket burns for 30 seconds with constant thrust. What is the thrust?

(continued on next page)

L2 ×

Physics 10424 Exam 1 February 10, 2012

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IV. An incompressible fluid with density ρ flows into a cylindrical pipe with sections of varying radii and across a change in height d. At each section, a column of the fluid acting as a barometer with the top open to the atmosphere is inserted. The input fluid speed is v, and the fluid flows out of the end of the pipe at D. Once the pipe narrows to radius R, the radius remains constant. The pipe continues to the left. (a) Find the speed of the fluid at D in terms of v. (b)-(d) Find the heights of the fluids in each of the (static) barometric columns (hA, hB, hC), measured from the centerline of the flow at each point. You may assume you know atmospheric pressure, Patm. Hint: You know the pressure at D. Use this and work backwards.

V. A strange bowl has a shape such that the radius increases with the fourth root of the height, ! =   !ℎ + !!!

! , where κ is an appropriate constant. The bowl has a height H, and the opening in the bottom has a (small-ish) radius r0. The hole in the bottom is stopped up, and the bowl is filled to the top with water. At time t = 0, the stopper is removed, and the water level in the bowl begins to drop. Do not make the approximation that r >> r0, i.e., take the velocity of the fluid in the bowl into consideration.

a) Write a version of Bernoulli’s equation relating fluid flow quantities where the fluid in the bowl has a height h to the water flowing out of the bottom.

b) Using the continuity relation, how is the speed at which the water level is decreasing in the bowl related to the speed of water flowing out of the bottom?

c) Find the speed at which the water level is decreasing in the bowl.

d) How long does it take the water to drain out? (Note: no calculus should be necessary here.)

6R 3R

R

hA hB

hC

D

C

A

B

d

r

h

r0

H

h = 0

v