Physics 4A Winter 2016 Test 3

Name:

March 4, 2016

Please show your work! Answer as many questions as you can, in any order. Calculatorsare allowed. Books and notes are not allowed. Use any blank space to answer questions, butplease make sure it is clear which question your answer refers to.

For all questions that require answers that are algebraic expressions, the expressions mustbe only in terms of the variables given in the question, and universal constants, such as g.

g = 9.8 ms−2 (If you like you can use g = 10 ms−2, but make your choice clear.)

Equations

vf = vi + ve ln(mi

mf

)Fthrust =

∣∣ve dmdt

∣∣Moments of Inertia

All objects listed here have mass M .

Thin rod, length L, axis through CM perpendicular to rod: I = 112ML2

Thin rod, length L, axis through endpoint of rod, perpendicular to rod: I = 13ML2

Sphere, radius R, axis through CM: I = 25MR2

Cylinder or disc, radius R, axis through CM: I = 12MR2

Thin ring, radius R, axis through CM: I = MR2

Trigonometric Identities

sin2 θ + cos2 θ = 1sin(2θ) = 2 sin(θ) cos(θ).cos(2θ) = cos2 θ − sin2 θsin(α± β) = sinα cos β ± cosα sin βcos(α± β) = cosα cos β ∓ sinα sin βcosα cos β = 1

2[cos(α− β) + cos(α + β)]

sinα sin β = 12[cos(α− β) − cos(α + β)]

sinα cos β = 12[sin(α + β) + sin(α− β)]

sin(θ + π

2

)= cos θ

cos(θ + π

2

)= − sin θ

sec θ := 1cos θ

csc θ := 1sin θ

cot θ := 1tan θ

1

1. A girl of mass mg is standing on a plank of mass mp. Both are originally at rest ona frozen lake that constitutes a frictionless, flat surface. The girl begins to walk alongthe plank at a constant velocity vgp to the right relative to the plank. (The subscriptgp denotes the girl relative to plank.)

(a) What is the magnitude of the velocity vpi of the plank relative to the surface ofthe ice? [4 pts]

(b) What is the magnitude of the girl’s velocity vgi relative to the ice surface? [4 pts]

(c) If the girl walks a distance x along the plank and then stops, how far does theplank move relative to the surface of the ice? [3 pts]

(d) How far does the center-of-mass of the girl-plank system move relative to theice? [2 pts]

2

2. Two shuffleboard disks of equal mass, one orange and the other yellow, are involved inan elastic, glancing collision. The yellow disk is initially at rest and is struck by theorange disk moving with a speed vi. After the collision, the orange disk moves along adirection that makes an angle θ with its initial direction of motion. The velocities ofthe two disks are perpendicular after the collision.

(a) Considering momentum, determine the final speed of each disk. [7 pts]

(b) Confirm that your final speeds are consistent with an elastic collision. [3 pts]

(c) Could the two disks final velocities be perpendicular if the collision was inelas-tic? [1 pt]

(d) Assume the two disks are solid and their internal energies do not change. Couldthe angle between the two disks’ final velocities be greater than 90◦? What wouldhappen to the kinetic energy in that hypothetical collision? [2 pts]

3

3. An object with a mass of m = 6.00 kg is attached to the free end of a light stringwrapped around a reel of moment of inertia I = 0.120 kg m2 and radius R = 0.300 m.The reel is free to rotate in a vertical plane about the horizontal axis passing throughits center as shown. The suspended object is released from rest.

Problems 329

and the pulley is a hollow cylinder with a mass of M 5 0.350  kg, an inner radius of R1 5 0.020 0  m, and an outer radius of R2 5 0.030 0 m. Assume the mass of the spokes is negligible. The coefficient of kinetic friction between the block and the horizontal surface is mk 5 0.250. The pulley turns without friction on its axle. The light cord does not stretch and does not slip on the pul-ley. The block has a velocity of vi 5 0.820 m/s toward the pulley when it passes a reference point on the table. (a) Use energy methods to predict its speed after it has moved to a second point, 0.700 m away. (b) Find the angular speed of the pulley at the same moment.

54. Review. A thin, cylindri-cal rod , 5 24.0  cm long with mass m 5 1.20 kg has a ball of diameter d 5 8.00  cm and mass M 5 2.00 kg attached to one end. The arrangement is originally vertical and stationary, with the ball at the top as shown in Figure P10.54. The com-bination is free to pivot about the bottom end of the rod after being given a slight nudge. (a) After the combination rotates through 90 degrees, what is its rotational kinetic energy? (b) What is the angular speed of the rod and ball? (c) What is the linear speed of the center of mass of the ball? (d) How does it compare with the speed had the ball fallen freely through the same distance of 28 cm?

55. Review. An object with a mass of m 5 5.10 kg is attached to the free end of a light string wrapped around a reel of radius R 5 0.250  m and mass M 5 3.00 kg. The reel is a solid disk, free to rotate in a ver-tical plane about the horizontal axis passing through its center as shown in Figure  P10.55. The suspended object is released from rest 6.00  m above the floor. Determine (a) the tension in the string, (b) the accel-eration of the object, and (c) the speed with which the object hits the floor. (d)  Verify your answer to part (c) by using the isolated system (energy) model.

d

m!

M

Figure P10.54

AMT

M

model the hands as long, thin rods rotated about one end. Assume the hour and minute hands are rotating at a constant rate of one revolution per 12 hours and 60 minutes, respectively.)

50. Consider two objects with m1 . m2 connected by a light string that passes over a pulley having a moment of inertia of I about its axis of rotation as shown in Figure P10.50. The string does not slip on the pulley or stretch. The pulley turns without fric-tion. The two objects are released from rest separated by a vertical distance 2h. (a) Use the principle of conservation of energy to find the translational speeds of the objects as they pass each other. (b) Find the angular speed of the pul-ley at this time.

51. The top in Figure P10.51 has a moment of inertia of 4.00 3 1024 kg ? m2 and is initially at rest. It is free to rotate about the stationary axis AA9. A string, wrapped around a peg along the axis of the top, is pulled in such a manner as to maintain a constant tension of 5.57 N. If the string does not slip while it is unwound from the peg, what is the angular speed of the top after 80.0 cm of string has been pulled off the peg?

52. Why is the following situation impossible? In a large city with an air-pollution problem, a bus has no combustion engine. It runs over its citywide route on energy drawn from a large, rapidly rotating fly-wheel under the floor of the bus. The flywheel is spun up to its maximum rotation rate of 3 000 rev/min by an electric motor at the bus terminal. Every time the bus speeds up, the flywheel slows down slightly. The bus is equipped with regenerative braking so that the flywheel can speed up when the bus slows down. The flywheel is a uniform solid cylinder with mass 1 200 kg and radius 0.500 m. The bus body does work against air resistance and rolling resistance at the average rate of 25.0 hp as it travels its route with an average speed of 35.0 km/h.

53. In Figure P10.53, the hanging object has a mass of m1 5 0.420 kg; the sliding block has a mass of m2 5 0.850 kg;

2h

I

m1

m2

R

Figure P10.50

A

A!

FS

Figure P10.51

R2 R1

m1

m2

Figure P10.53

M

R

m

Figure P10.55(a) Determine the tension in the string and [4 pts]

(b) the angular acceleration of the wheel. [4 pts]

(c) What is the angular displacement of the reel when the object has descendedthrough a distance of 1.00 m? [2 pts]

(d) What is the angular speed of the reel at that moment? [3 pts]

4

4. A thin, cylindrical rod, ` = 20.0 cm long with mass m = 1.10 kg has a ball of diameterd = 4.00 cm and mass M = 2.00 kg attached to one end. The arrangement is originallyvertical and stationary, with the ball at the top as shown. The combination is free topivot about the bottom end of the rod after being given a slight nudge.

Problems 329

and the pulley is a hollow cylinder with a mass of M 5 0.350  kg, an inner radius of R1 5 0.020 0  m, and an outer radius of R2 5 0.030 0 m. Assume the mass of the spokes is negligible. The coefficient of kinetic friction between the block and the horizontal surface is mk 5 0.250. The pulley turns without friction on its axle. The light cord does not stretch and does not slip on the pul-ley. The block has a velocity of vi 5 0.820 m/s toward the pulley when it passes a reference point on the table. (a) Use energy methods to predict its speed after it has moved to a second point, 0.700 m away. (b) Find the angular speed of the pulley at the same moment.

54. Review. A thin, cylindri-cal rod , 5 24.0  cm long with mass m 5 1.20 kg has a ball of diameter d 5 8.00  cm and mass M 5 2.00 kg attached to one end. The arrangement is originally vertical and stationary, with the ball at the top as shown in Figure P10.54. The com-bination is free to pivot about the bottom end of the rod after being given a slight nudge. (a) After the combination rotates through 90 degrees, what is its rotational kinetic energy? (b) What is the angular speed of the rod and ball? (c) What is the linear speed of the center of mass of the ball? (d) How does it compare with the speed had the ball fallen freely through the same distance of 28 cm?

55. Review. An object with a mass of m 5 5.10 kg is attached to the free end of a light string wrapped around a reel of radius R 5 0.250  m and mass M 5 3.00 kg. The reel is a solid disk, free to rotate in a ver-tical plane about the horizontal axis passing through its center as shown in Figure  P10.55. The suspended object is released from rest 6.00  m above the floor. Determine (a) the tension in the string, (b) the accel-eration of the object, and (c) the speed with which the object hits the floor. (d)  Verify your answer to part (c) by using the isolated system (energy) model.

d

m!

M

Figure P10.54

AMT

M

model the hands as long, thin rods rotated about one end. Assume the hour and minute hands are rotating at a constant rate of one revolution per 12 hours and 60 minutes, respectively.)

50. Consider two objects with m1 . m2 connected by a light string that passes over a pulley having a moment of inertia of I about its axis of rotation as shown in Figure P10.50. The string does not slip on the pulley or stretch. The pulley turns without fric-tion. The two objects are released from rest separated by a vertical distance 2h. (a) Use the principle of conservation of energy to find the translational speeds of the objects as they pass each other. (b) Find the angular speed of the pul-ley at this time.

51. The top in Figure P10.51 has a moment of inertia of 4.00 3 1024 kg ? m2 and is initially at rest. It is free to rotate about the stationary axis AA9. A string, wrapped around a peg along the axis of the top, is pulled in such a manner as to maintain a constant tension of 5.57 N. If the string does not slip while it is unwound from the peg, what is the angular speed of the top after 80.0 cm of string has been pulled off the peg?

52. Why is the following situation impossible? In a large city with an air-pollution problem, a bus has no combustion engine. It runs over its citywide route on energy drawn from a large, rapidly rotating fly-wheel under the floor of the bus. The flywheel is spun up to its maximum rotation rate of 3 000 rev/min by an electric motor at the bus terminal. Every time the bus speeds up, the flywheel slows down slightly. The bus is equipped with regenerative braking so that the flywheel can speed up when the bus slows down. The flywheel is a uniform solid cylinder with mass 1 200 kg and radius 0.500 m. The bus body does work against air resistance and rolling resistance at the average rate of 25.0 hp as it travels its route with an average speed of 35.0 km/h.

53. In Figure P10.53, the hanging object has a mass of m1 5 0.420 kg; the sliding block has a mass of m2 5 0.850 kg;

2h

I

m1

m2

R

Figure P10.50

A

A!

FS

Figure P10.51

R2 R1

m1

m2

Figure P10.53

M

R

m

Figure P10.55

(a) What is the moment of inertia of the rod-and-ball system about the pivot? [4 pts]

At the instant the rod is horizontal, find

(b) the rotational kinetic energy of the system and [3 pts]

(c) the angular speed of the rod and ball. [3 pts]

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