Physics 1301 Review September 2010

This is a collection of problems which covers the range of topics found in this course. Many of the problems have parts that deal with a single situation, while others have parts which are only related to the same topic. These problems are frequently longer or somewhat more difficult than those found on final examinations, but they are indicative of the sorts of physical analysis and calculations you may be expected to carry out. They are intended to provide an extensive review of the course material. Keep in mind that no list such as this one will cover everything you have studied in the course; you should be sure to review your lecture notes and homework thoroughly. Many of the problems in this Review are based on ones appearing in Fishbane, Gasiorowicz, and Thorton, Physics for Scientists and Engineers, 3rd edition [FGT3] and Halliday, Resnick, and Walker, Fundamentals of Physics, 6th edition [HRW6] and its Problem Supplement #1 [HRW6s]. Answers are provided following this list. A solution set is presented separately. 1. Estimate the amount of water that one person drinks in their lifetime. Suggest a body of water of comparable volume. Some population analysts estimate that about 300 billion people may have lived on Earth over historical time. What is the total amount of water that has been drunk by Humankind? If that water has been well mixed in the oceans to a depth of 100 meters, what fraction of that oceanic water have people used? How many molecules of water in the next liter you drink have been shared by another person? 2. The flow rate of liquid through a cylindrical tube was studied by G.H.L. Hagen, who experimentally derived a formula describing it in 1838. The relationship was formulated by J.-L.M. Poiseuille in 1840 and is known usually as Poiseuille’s Law (but also as the Hagen-Poiseuille Law). This equation relates the volume rate of flow through the tube, Q = dV/dt , in terms of the radius of the tube, r ; the length of the tube, L ; the difference of pressure between the ends of the tube, ΔP ; and the resistance of the liquid to flow, its viscosity, η , which has units of (force · time)/area . Use dimensional analysis to find the form of this Law, without its dimensionless numerical constants. You will need one piece of empirical information to resolve the relationship fully: doubling the length of the tube reduces the flow rate by one-half. 3. While we have frequently treated it as such, we know that the acceleration due to Earth’s gravity near its surface is not constant. We can improve upon this by using the linear approximation a(r) = −( GM/R

e

2 ) · [ 3 − (2r/ Re) ] , r ≥ R

e , which is accurate to 1%

up to radii of about 6800 km. (The Earth’s average radius is Re ≈ 6380 km.)

Find the speed of a falling object dropped from a height h0 as a function of height.

Also find the speed as a function of time. (We will neglect atmospheric drag here.) Compare these results to those for constant g = ( GM/R

e

2 ) . [based on FGT3 2-68]

4. A freight train travelling at 135 km/hr rounds a bend. As the train enters a straightaway, the station ahead comes into view and the engineer abruptly notices that a passenger train is stopped on the same track the freight train is using; the end of that train is 3000 meters away. He immediately applies the brakes to avoid a collision. What deceleration must the brakes be able to provide in order to do so? In fact, these brakes can only achieve a deceleration of a

1 = −0.16 m/sec2 .

Meanwhile, at the station, the alert engineer of the passenger train spots the advancing freight and reckons that it will not be able to stop in time. Reacting quickly, she is able to get the passenger train accelerating just 30 seconds after the freight engineer applies his brakes. However, the passenger train is only able to accelerate at a

2 = 0.12 m/sec2 .

Has she successfully averted a collision between the trains? If so, what is the closest approach made between the ends of the trains, how fast are they moving at that moment, and how far from the station does that happen? If there is a collision, where does it occur and what is the relative speed of the trains at that moment? [based on HRW6 2-34, -38] 5. A hotel elevator can travel 78 meters along its shaft. On a long run, the car requires 5 seconds to accelerate its cruising speed and 8 seconds to decelerate to a stop. It is found that on a run covering the entire shaft, the car reaches the half-way point in a climb in 47.5% of the total travel time. How long does the trip take? What are the values of the two accelerations and the cruising speed? Now consider a child passenger tossing a ball straight up inside the car while it is in motion. The ball is thrown from a height of 0.5 meters above the floor and stops just short of the ceiling before being caught at the same height. The interior of the car is 2.5 meters high. What initial velocity must the child give the ball and how long is it in flight? Answer these questions for each phase of the elevator’s travel. [based on HRW6 2-61] Suppose a Health-O-Meter scale were placed in the elevator and an adult of 800 N weight (as measured, say, in the hotel lobby) chose to weigh themselves while riding the elevator. What would the scale read during each phase of travel? 6. You are sitting in a location with a view of a baseball stadium; however, the game is blocked from view, as you are at eye-level with the top of the wall, which runs parallel to the ground. You see a struck ball emerge into view, reach the apex of its flight at a height twice that of the wall, and become lost to view behind the wall again 3.0 seconds later. The points where the ball appeared and disappeared are separated by 70 meters. Suppose the ball’s line of flight was parallel to the wall and the ball was struck and caught one meter above the ground. Suppose further that air resistance was negligible (not quite realistic). What was the initial velocity of the ball? How far did the ball travel horizontally between batter and outfielder? How high is the wall? [based on HRW6s 2-106] 7. An adult and child decide to have a game of catch at a large park, but the largest suitable space is on a 15º slope. They stand 20 meters apart, as measured along the slope, with the child taking the uphill position. If both players choose a comfortable throwing angle of 30º relative to the slope, how fast much each person throw the ball to reach the other one? If each player uses this throwing angle and can throw as fast as 20 m/sec, how far along the slope could the ball land in each direction?

8. A 12 kg. ladder with legs 2.8 m. long can be opened out to stand up, with the arrangement secured by a tie-rod of length 0.9 m. located halfway up the legs. A worker weighing 900 N climbs the ladder to stand on a rung 2.3 m. measured along one leg from its foot. The floor is taken to be frictionless. What are the tension in the tie-rod and the forces on the ladder from the floor at each pair of its feet? Suppose the tie-rod has been damaged and replaced (improperly) by an equal length of material with a “crushing strength” of 200 N. What is the highest safe point measured along the legs to which this same worker may now climb? [based on HRW6 13-31] 9. a) A set of four uniform bricks (or blocks or books) of identical mass and length L are stacked so that they overhang one another as far as possible, yet still maintain static equilibrium (although not very stably). What are the respective distances of overhang (and how far does the end of the top brick extend beyond the end of the table)?

b) Now we rearrange the bricks into the new configuration shown (this illustrates the principle of the cantilever). For static equilibrium, find the distances marked.

[based on HRW6 13-24, -34] 10. a) What amount of force F must be applied horizontally at the axle of a bicycle wheel of radius R and mass M in order to raise it over a step of height h ? (Treat the wheel as a uniform hoop.) Does the answer change if the wheel is replaced by a solid uniform disk of identical mass and radius? [based on HRW6 13-21, FGT3 10-70]

b) A uniform cube of side L rests on a horizontal floor. The coefficient of static friction between cube and floor is µ

s . A horizontal push of force F is applied at the

midline of one of the upright sides of the cube at a height h above the floor. At what height will the box tip over rather than slide across the floor? [HRW6s 13-53, FGT 11-72] c) A stack of four such cubes is lashed together and placed upright in the cargo space of a flatbed truck. The coefficient of static friction between the bottommost cube and the floor of the bed is µ

s = 0.32 . What is the greatest magnitude of acceleration the

truck may have in order to keep the stack from toppling over? [based on FGT3 11-74] 11. Two metal spheres are placed in a closed-bottomed cylindrical tube of radius 4 cm.

One is made of aluminum (density 2700 kg/m3) and has a radius of 2.5 cm.; the other is

pure copper (density 9000 kg/m3) with a radius of 3.0 cm. The copper ball has been placed on top of the aluminum one. Once in static equilibrium, what are the magnitudes of each of the forces on the two balls? (You may express the forces in terms of mg , the weight of the aluminum ball.) [based on FGT3 11-79] 12.

a) A common variation of the Atwood machine is to use a wedge and arrange the blocks and pulley so that each block slides on individual surfaces of the wedge. Here we have one block with mass m

1 = 1.8 kg. on the inclined surface 40º above the horizontal;

the other block with mass m2 = 3.0 kg moves on the other inclined surface. All of the

surfaces are frictionless, as is the pulley. What angle θ2 above the horizontal must the

second surface have in order for the system to be in static equilibrium? b) Suppose now that the surfaces have a coefficient of static friction µ

s = 0.38 and

that θ2 = 32º ; m

1 is unchanged. What range of masses may be used for m

2 so that the

system remains in static equilibrium? c) If we now choose to use masses of m

1 = 1.8 kg. and m

2 = 9.2 kg, and the surfaces

of the inclines in part (b) have a coefficient of kinetic friction µk = 0.26 , what is the

acceleration of the masses? d) For the same arrangement as in part (c), the pulley is no longer frictionless, but can rotate. If the observed acceleration of the masses is 0.88 m/sec2 , what is the moment of inertia of the pulley, given that its radius is 12.5 cm.? What are the tensions in the two lengths of cord? [based on FGT3 5-13, -14, 10-50, -56, HRW6s 11-93]

13.

a) The belt of a horizontal conveyor moves with a constant speed of 3.0 m/sec. A small graphite block is dropped straight down onto the belt. This causes the block to skid along the surface of the belt until it is matches speed with the belt. If the coefficients of friction between block and belt are µ

k = 0.2 and

µs = 0.3 , how long is the skid mark left by the graphite?

[FGT3 5-89]

b) A block of mass m

1 = 0.7 kg. rests on top of a larger block of mass m

2 = 1.3 kg.,

which starts at rest on a frictionless tabletop. A massless cord is attached to m2, which

is used to apply a horizontal force of 6 N. If the contact between the two blocks is frictionless, what are the accelerations of each block? c) If instead the coefficients of friction between the two blocks are µ

s = 0.45 and µ

k =

0.32, what are the accelerations of the blocks for the same 6 N applied force? What is the largest force that can be applied so that the two blocks move together? What are the accelerations of the blocks if the applied force is 10 N? [based on FGT3 5-36] d) Using the blocks in part (c), we now replace the cord with a spring with constant k = 72 N/m to the lower block. What is the largest amplitude with which the blocks may oscillate so that they will continue to move together? [based on HRW6 16-16] e) A popular magician’s trick is to yank the tablecloth out from between a dinner table and the place settings of plates, glasses and cutlery, without sending everything crashing to the floor; in fact, when performed properly, the items on the table move very little! Suppose, for convenience, that the coefficient of kinetic friction between all the tableware and the tablecloth is µ

k = 0.15 and that between the tableware and the bared

table surface is µk’ = 0.24. We are going to perform this trick at a round table of 1.07 m

(42”) diameter, where no item may move more than 8 cm. (3”) before something will fall off the table. Within what time must we be prepared to pull the tablecloth free of the table to avoid causing anything else to be pulled off with it? 14. A plumb bob is dangling from the roof light inside a car on a level street which is turning a corner by following a circular arc of radius 27 ft. while moving at 11 mph. How far off-vertical is the plumb bob during the turn? Now the bob is hanging from the ceiling of the lobby of Tate Labs at the University of Minnesota - Twin Cities (latitude 45º N.). Let us take the Earth’s gravity field to be strictly radial here. The bob hangs “vertically”, but due to the Earth’s rotation, it is not aligned radially. Find the size of the angle of deviation.

15. a) Raindrops typically fall about 3 km. through the open air upon leaving the cloud. How fast would they strike the ground, were there no air resistance? In fact, we notice that raindrops reach the ground far more slowly. Suppose that: 1) the air drag on them is proportional to v2 and to cross-sectional area; 2) that all raindrops have the same drag coefficient; and 3) that the drops remain spherical during their descent (which isn’t really true). Compare the radii of droplets which land at 1 m/sec with those which land at 5 m/sec. If the droplets grow within a cloud such that surface area is proportional to time, what is the relative amount of time that these droplets have spent within their originating cloud? b) Now consider hailstones, which we will take to all have the same density and drag coefficient. They are kept aloft in the cloud by wind drag due to turbulence, where they grow until their weights are too great to support. If we assume the same velocity law for drag here as for the raindrops, how does the maximum wind speed compare in a cloud dropping baseball-sized hail (7.5 cm. diameter) to that in a cloud dropping pea-sized hail (0.75 cm. diameter)? 16. a) The engine of a 900 kg. automobile is generating 95 kW (127 hp) of useful mechanical power in order to sustain its speed at 65 mph during interstate highway driving. If the coefficient of rolling friction is µ = 0.025 and the car is on level pavement, how much power is being used to overcome air drag? b) If the drag on the automobile is proportional to v2 , but the frictional force remains the same, what power must the engine supply to sustain the car at 80 mph? c) If the engine is held at this higher power level and the car is now taken up a 12% grade (a climb of 12 m. for every 100 m. traveled horizontally), at what speed would the car now move? Assume the rolling friction is essentially unchanged. d) With the engine off and the gearing in neutral, at what speed would the car coast on a 12% downgrade? (Neglect internal friction in the transmission and wheels.) [based on HRW6 9-56, -57, -58, HRW6s 9-70] 17.

A puck of mass 0.3 kg. glides in a circle of radius 45 cm. on a frictionless table surface. It is maintained in this motion by the tension in a cord to which it is fastened; the cord runs through a hole in the center of the table and is attached at its other end to a 1.3 kg. hanging mass. At what tangential speed must the puck move in order to keep the hanging mass stationary? [based on HRW6 6-41] Now suppose the puck is moving on an air table with a coefficient of kinetic friction of µ

k = 0.0008. At what rate is the hanging mass descending at the moment described

above? Is the puck speeding up or slowing down as it spirals toward the hole?

18. A marble is placed at the North Pole of a globe in a supporting stand and released from rest; the marble then begins to roll without slipping along a meridian. At what latitude on the globe does the marble leave the globe’s surface? Would it matter if a smooth finger ring were used instead? (Assume these objects have uniform densities.) [based on HRW6 8-35] 19. This is admittedly a goulash problem in mechanics.

a) A block of mass m

1 = 0.100 kg. is initially held at rest in a spring launcher; the

spring has constant k1 = 100 N/m and is initially compressed by Δx = 11.0 cm. The

launcher sits on a frictionless incline. It is located at a height H = 35 cm. above the bottom of a frictionless concave surface and is aimed at an angle θ

1 = 25º below the

horizontal. When this block is launched, it races down the curved ramp. At the bottom, it collides elastically with a second block of mass m

2 . The second block now starts up the

other incline. However, at a height h1 = 2.5 cm. above the bottom, the incline ceases to

be curved or frictionless: it is now straight and inclined θ2 = 9º above the horizontal and

has a coefficient of kinetic friction of µk = 0.17 . After ascending by the additional

height h2 = 5 cm. , this block strikes a spring bumper with constant k

2 = 25 N/m , which

brings the block momentarily to rest after being compressed by Δy = 5.0 cm . Meanwhile, on the other side of this set-up, the first block has traveled back up its own ramp, where it just comes to rest momentarily as it reaches the spring launcher. What must the mass m

2 be?

b) On a second launch, we make one slight change by altering the compression Δx of the launcher spring. Now the second block travels up the rough incline and just comes to rest upon reaching the spring bumper, without compressing it. What is Δx this

time? Will the second block m2 slide back down the ramp?

20. A spacecraft is instructed remotely to perform a trajectory correction maneuver in interplanetary space. At that moment, its total mass including fuel is 2750 kg. The onboard chemical rocket engine causes 0.26 kg/sec of exhaust to be expelled through its nozzle at a speed of 4000 m/sec relative to the spacecraft. For this maneuver, the engine is to be fired for 18 seconds. How much mass is expelled? What is the velocity change of the spacecraft and its average acceleration during the engine firing? Suppose the spacecraft (taken to have identical initial mass) is instead equipped with a xenon-ion thruster which expels 5⋅1018 ions per second at a speed of 2⋅105 m/sec.

(Xenon has an atomic weight of 131 amu, where 1 amu = 1.66⋅10-27 kg.) To achieve the same change in velocity, how much mass must the spacecraft now expel? How long should the thruster be operated? What is the average acceleration of the spacecraft?

21. a) A thin uniform rod of mass 0.75 kg. and length 1.2 m. is initially at rest. It is then struck at one end by a small projectile of mass 0.015 kg., which was moving at a speed of 120 m/sec at an angle of 45º to the rod before ricocheting off it. After this collision, how far will the center of mass move by the time the rod has completed 10 revolutions? [based on FGT3 9-76, HRW6 12-50] b) A tall cylindrical smokestack topples over when its base is ruptured for demolition. We may treat it as a thin, uniform, rigid rod of length H; let θ be the angle that the smokestack is tilted from the vertical. Expressed as functions of θ , what are the angular speed of the chimney and the radial and tangential accelerations at its top? At what angle θ does the tangential acceleration reach g ? [HRW6 11-67] 22. a) A disk of rotational inertia I , mass M , and radius R is released from rest on a plane inclined by angle θ to the horizontal. The coefficient of static friction between the

disk and the plane surfaces is µs . What is the critical angle θ

c below which the disk can

roll without sliding? [HRW6s 12-83] b) A bowler throws a bowling ball of radius 11 cm. down a lane. Initially it slides on the polished wood surface with linear speed 7.5 m/sec and zero rotational velocity. Kinetic friction will act to reduce the linear speed while producing a torque accelerating the speed of rotation; the coefficient of kinetic friction is µ

k = 0.31 . How far will the

ball slide before beginning to roll? How long will this take? Once rolling commences, what are the linear and rotational speeds of the bowling ball? [HRW6 12-14] c) A disk with radius R

1 and rotational inertia I

1 about its center spins with angular

velocity ω0 . It is placed into contact edge-to-edge with a second disk of radius R2 and

rotational inertia I2 about its center; this disk is initially at rest. Both disks turn on axes

perpendicular to the plane containing them. The spinning disk imparts rotation to the stationary disk through friction at their rims until a steady state is reached. What are the final angular velocities of the two disks? (Note: because this change was accomplished by a dissipative force -- friction -- it is not the case that kinetic energy or angular momentum are conserved. It is necessary to analyze the behavior of the forces involved.) [HRW6 12-37]

23. a) An old-fashioned escapement clock (a “grandfather clock”) is regulated by the oscillation of a heavy pendulum set in motion. The pendulum is a long heavy shaft with a slot at the lower end, in which the position of a massive “bob” may be adjusted. For our particular clock, the shaft is essentially uniform in density and has a length L = 1.100 m. and mass m = 2.000 kg. ; the bob is basically a uniform disk of radius r = 0.080 m. and mass M = 5.000 kg. (We will treat the pivot as being located just at the upper end of the shaft.) This physical pendulum should be adjusted so that its period is 2.000 seconds. (The escapement allows the clock gearing to advance at each half-oscillation.) How far is the center of the bob from the pivot point? Let us suppose that this clock is originally adjusted for a location at sea level on the Equator (where g

e = 9.78033 m/sec2). [based on FGT3 13-64]

b) With the passing of the seasons, we find that, while the pendulum is adjusted correctly for the warmer months, with the arrival of cooler weather, the length of the shaft has contracted by 0.10% . How much time will our clock be off by after one week? In what way and by how much should the bob be adjusted? c) Consider instead that our clock is transported to Auckland, New Zealand at 37º South latitude and at sea level, and at a temperature similar to that in part (a). Keep in mind that the effective value of g depends upon latitude. Making comparisons to part (a) then, answer the questions of part (b). d) As another alternative, our clock remains near the Equator but is now delivered to a location in Ecuador at 3500 m. above sea level. Again making comparisons to part (a), answer the questions of part (b). You may use the linear approximation for the value of g provided in problem 3 above. [based on HRW6s 16-71, -72, -90] 24. A uniform solid cylinder of mass M = 6.4 kg. and radius R = 7.0 cm. is fitted with a frictionless axle along the cylinder’s long axis. A spring of constant k = 4.5 N/m is attached to a bracket connected to the axle. The cylinder is now perturbed from the equilibrium position of the spring and rolls without slipping on the horizontal table surface. What is the period of oscillation of this device?

Now we will consider the axle to actually have friction, creating a small resistive force −0.220 · (dθ/dt) N on the cylinder. What is the period of oscillation in this case? [based on FGT3 13-99 and HRW6s 16-99] Answers 1. ~40,000 liters (small pond); ~1 · 1013 m3 ; ~3 · 10-4 ; ~1·1022 (all are broad estimates)

2. Q ∝ ( r4 • ΔP )/( η • L )

3.

€

v(h) = 2g ⋅ (h0 − h ) −(h02 − h2)RE

⎡

⎣ ⎢

⎤

⎦ ⎥

⎧ ⎨ ⎪

⎩ ⎪

⎫ ⎬ ⎪

⎭ ⎪ 1/ 2 ;

€

v(t) = − gt ⋅ 1− 13gRE

t 2⎛

⎝ ⎜

⎞

⎠ ⎟

4. −0.234 m/sec2 ; collison is averted: 38 m. (closest approach), 14.0 m/sec , stopped train has moved forward 820 m. 5. 30 sec. ; a

1 = 0.664 m/sec2 , a

2 = −0.415 m/sec2 ; v = 3.32 m/sec

at a1: 6.47 m/sec , 1.235 sec. , 854 N ; at v : 6.26 m/sec , 1.277 sec. , 800 N ;

at a2: 6.13 m/sec , 1.305 sec. , 766 N

6. 30.9 m/sec at 41.1º above the horizontal; 96.8 m. ; 11.0 m. 7. child throwing to adult: 13.8 m/sec , 42.2 m. ; adult throwing to child: 16.1 m/sec , 30.9 m. 8. tension in tie-rod: 271 N in compression; normal force at rear feet: 589 N , at forward feet: 429 N ; 1.65 m. 9. a) a

1 = (1/2)L , a

2 = (1/4)L , a

3 = (1/6)L , a

4 = (1/8)L , h = (25/24)L

b) b1 = (3/5)L , b

2 = (1/2)L , h = (11/10)L

10. a) F = [ ( 2Rh − h2 )1/2/(R−h) ] · Mg, regardless of rotational inertia of wheel

b) hmax

= ( mg/F )·( L/2 ) c) amax

= (1/4) g 11. contact force with walls: N

l = N

r ≈ 2.939 mg ;

contact force between spheres: Nc ≈ 6.467 mg ;

contact force with bottom of tube: Nv = (169/25)·mg = 6.76 mg

12. a) 22.7º b) 0.743 ≤ m2 ≤ 8.10 kg. c) m

2 slides downhill at 1.19 m/sec2

d) 0.0602 kg·m2 ; T1 = 16.45 N , T

2 = 19.83 N

13. a) 2.29 m. b) a1 = 0, a

2 = 4.62 m/sec2

c) a1 = a

2 = 3.00 m/sec2 ; F

max = 8.83 N ; a

1 = 3.14 m/sec2, a

2 = 6.00 m/sec2

d) A = 0.123 m. e) 0.26 sec. 14. 16.7º ; 0.099º 15. a) 240 m/sec ; 1 : 25 ; 1 : 625 b) 3.2 : 1 16. a) 88.6 kW b) 171.6 kW c) 33.0 m/sec ( 74 mph ) d) 15.2 m/sec ( 34 mph ) 17. 4.37 m/sec ; 0.54 mm/sec , puck slows down as it spirals in toward the hole 18. marble: 36.0º N ; ring: 30.0º N.

19. a) m2 = 0.402 kg. b) Δx = 9.55 cm. ; no

20. chemical rocket: Δm = 4.68 kg. , Δv = 6.81 m/sec , aave

= 0.379 m/sec2

ion engine: Δm = 0.094 kg. , Δt = 85,900 sec , aave

= 7.9·10-5 m/sec2 21. a) 12.55 m. b) ω = [ (3g/H) · ( 1 − cos θ ) ]1/2 ; α

rad = 3g · ( 1 − cos θ ) ;

αtan

= (3/2) · g · sin θ ; θ = sin-1(2/3) ≈ 41.8º

22. a) tan θc ≤ µ

s · [ 1 + ( MR2/I

CM ) ]

b) 4.52 m. , 0.705 sec., 5.36 m/sec , 48.7 rad/sec c) ω

1 = ω

0 · ( I

1R

2

2 )/( I2R

1

2 + I1R

2

2 ) , ω2 = ω

0 · ( I

1R

1R

2 )/( I

2R

1

2 + I1R

2

2 ) 23. a) 1.0422 m. b) 306 sec. fast, lower bob by 1.2 mm. c) 396 sec. fast, lower bob by 1.5 mm. d) 317 sec. slow, raise bob by 1.2 mm. 24. 9.177 sec. ; 9.451 sec.

G. Ruffa -- revision 2/7/11