Physics 141, Final Exam

December 19, 2006, 4.00 pm - 7.00 pm

Do not turn the pages of the exam until you are instructed to do so.

Exam rules: You may use only a writing instrument and the equation sheet while taking this test. You may not consult any calculators, computers, books, nor each other.

1. Answer the multiple-choice questions (problems 1 – 10) by marking your answer on the scantron form. For each multiple-choice question (problems 1 – 10), select only one answer. Questions with more than one answer selected will be considered incorrect. If your student ID is not listed properly on the Scantron form, the form will not be processed and you loose points for all multiple-choice questions.

2. The analytical problems (11 – 17) must be answered in three blue exam booklets. You must answer problems 11 and 12 in blue booklet # 1, problems 13 and 14 in blue booklet # 2, and problems 15, 16, and 17 in blue booklet # 3. If you do not follow this convention there is no guarantee that the problems that appear in the wrong booklet will be graded.

3. The answer to each analytical problem must be well motivated and expressed in terms of the variables used in the problem. You will receive partial credit where appropriate, but only when we can read your solution. Answers that are not motivated will not receive any credit, even if correct.

4. At the end of the exam, you must hand in the blue exam booklets, the scantron form, and the exam. All items must be clearly labeled with your name and student ID number. If any of these items is missing, we will not grade your exam, and you will receive a score of 0 points.

5. You are not allowed to leave the exam room until 5.30 pm.

Name: __________________________________________________

ID number: ______________________________________________

Recitation Day/Time: ______________________________________

Useful Relations:

Moments of inertia of various objects of uniform composition.

Good luck!

Problem 1 (2.5 points)

A	B
C	D
E

Match the pictures of the ball parks shown above to the team names listed below.

1. Yankees

2. White Sox

3. Mets

4. Cubs

5. Red Sox

ABCDE =

1. 12345

2. 21453

3. 21534

4. 31542

5. 31254

6. 42351

Problem 2 (2.5 points)

The Figure on the right shows various possible trajectories of an object launched from the surface of the earth. What can you say about the total energy of the object for the various trajectories shown in the Figure?

1. E_{elliptical trajectory} > 0 J, E_{hyperbolic trajectory} = 0 J, and E_{parabolic trajectory} < 0 J.

2. E_{elliptical trajectory} > 0 J, E_{hyperbolic trajectory} < 0 J, and E_{parabolic trajectory} = 0 J.

3. E_{elliptical trajectory} = 0 J, E_{hyperbolic trajectory} > 0 J, and E_{parabolic trajectory} < 0 J.

4. E_{elliptical trajectory} = 0 J, E_{hyperbolic trajectory} < 0 J, and E_{parabolic trajectory} > 0 J.

5. E_{elliptical trajectory} < 0 J, E_{hyperbolic trajectory} = 0 J, and E_{parabolic trajectory} > 0 J.

6. E_{elliptical trajectory} < 0 J, E_{hyperbolic trajectory} > 0 J, and E_{parabolic trajectory} = 0 J.

Problem 3 (2.5 points)

Two satellites A and B of the same mass are going around Earth in concentric orbits. The distance of satellite B from Earth’s center is twice that of satellite A. What is the ratio of the centripetal force acting on B to that acting on A?

1. 1/8

2. 1/4

3. 1/2

4. 1/√2

5. 1

Problem 4 (2.5 points)

A sports car accelerates from zero to 30 mph in 1.5 s. How long does it take for it to accelerate from zero to 60 mph, assuming the power of the engine to be independent of velocity and neglecting friction?

1. 2 s

2. 3 s

3. 4.5 s

4. 6 s

5. 9 s

6. 12 s

Problem 5 (2.5 points)

When a small ball collides elastically with a more massive ball initially at rest, the massive ball tends to remain at rest, whereas the small ball bounces back at almost its original speed. Now consider a massive ball of inertial mass M moving at speed v and striking a small ball of inertial mass m initially at rest. The change in the small ball’s momentum is

1. mv

2. 2mv

3. Mv

4. 2Mv

5. none of the above

Problem 6 (2.5 points)

As you hold the string, a yoyo is released from rest so that gravity pulls it down, unwinding the string. What is the angular acceleration of the yoyo, in terms of the string radius R, the moment of inertia I, and the mass M?

1. gMR/(I + MR²)

2. g/R

3. g/(R + 2I/(MR))

4. gMR/I

Problem 7 (2.5 points)

The moment of inertia of a square plate of area 4R² and mass M, with respect to an axis through its center and perpendicular to the plate, is equal to (2/3)MR². A disk of radius R is removed from the center of the plate (see Figure). What is the moment of inertia of the remaining material with respect to the same axis?

1. (1/2)MR²

2. (2/3 - p/8)MR²

3. (1/6)MR²

4. (2/3 - p/4)MR²

Problem 8 (2.5 points)

Which of the following statements is false?

1. The buoyant force in liquid is much larger than the buoyant force in air.

2. The buoyant force is a result of small differences in molecular density of the air/liquid across the surface of the object.

3. The buoyant force is a result of differences in the average molecular velocity of the air/liquid molecules across the surface of the object.

4. The magnitude of the buoyant force increases with increasing temperature, due to the corresponding increase in the average molecular velocities.

Problem 9 (2.5 points)

A solid disk and a ring roll down an incline. The ring is slower than the disk if

1. m_ring = m_disk, where m is the mass.

2. r_ring = r_disk, where r is the radius.

3. m_ring = m_disk and r_ring = r_disk.

4. The ring is always slower, regardless of the relative values of m and r.

Problem 10 (2.5 points)

If all three collisions in the figure shown below are totally inelastic, which collisions cause(s) the most damage?

1. I.

2. II.

3. III.

4. I and II.

5. I and III.

6. II and III.

7. All three.

Problem 11 (25 points)

The operation of an automobile internal combustion engine can be approximated by a reversible cycle known as the Otto cycle, whose PV diagram is shown in the Figure below. The gas in cylinder at point a is compressed adiabatically to point b. Between point b and point c, heat is added to the gas, and the pressure increases at constant volume. During the power stroke, between point c and point d, the gas expands adiabatically. Between point d and point a, heat is removed from the system, and the pressure decreases at constant volume. Assume the gas is an ideal monatomic gas.

a. Assuming there are N molecules of gas in the system, what are the heats |Q_H| and |Q_L|? Express your answer in terms of N, k, T_a, T_b, T_c, and T_d.

b. What is the efficiency of the Otto cycle? Express your answer in terms of T_a, T_b, T_c, and T_d.

c. Express the efficiency of the Otto cycle in terms of just the compression ratio V_a/V_b and g.

d. How does the efficiency change when we replace the monatomic gas with a diatomic gas?

Problem 12 (25 points)

The Stanford Linear Accelerator Center (SLAC), located at Stanford University in Palo Alto, California, accelerates electrons through a vacuum tube of length L.

Electrons of mass m, which are initially at rest, are subjected to a continuous force F along the entire length of the tube and reach speeds very close to the speed of light.

a. Calculate the final energy of the electrons.

b. Calculate the final momentum of the electrons.

c. Calculate the final speed of the electrons.

d. Calculate the time required to travel the distance L.

Express all your answers in terms of the variables provided (m, F, and L) and the speed of light c.

Problem 13 (25 points)

A chain of metal links is coiled up in a tight ball on a frictionless table. The mass of the chain is M. You pull on a link at one end of the chain with a constant force F. Eventually the chain straightens out to its full length L, and you keep pulling until you have pulled your end of the chain a total distance d.

a. What is the speed of the chain at this instant?

b. When the chain straightens out, the links of the chain bang against each other, and their temperature rises. Calculate the increase in thermal energy of the chain, assuming that the process is so fast that there is insufficient time for the chain to lose much thermal energy to the table.

Express all your answers in terms of M, F, d, and L.

Problem 14 (25 points)

A satellite has four low-mass solar panels sticking out as shown in the Figure below. The satellite can be considered to be a uniform solid sphere of mass M and radius R. Initially the satellite is traveling to the right with speed v and rotating clockwise with angular speed w_i.

A tiny meteor of mass m, traveling at speed v₁, rips through one of the solar panels and continues in the same direction, but at reduced speed v₂.

a. Calculate the x and y components of the motion of velocity of the center of mass of the satellite after it is hit by the meteor. The positive x direction is taken to be the direction of the original motion of the satellite. The positive y direction is the direction from the center of mass of the satellite towards the solar panel that is hit by the meteor.

b. What is the angular velocity of the satellite after it is hit by the meteor? Specify both magnitude and direction.

Express all your answers in terms of m, M, h, R, q, v, v₁, v₂, and w_i.

Problem 15 (25 points)

A simple model of the Earth assumes that the density is uniform throughout its interior.

a. If the total mass of the Earth is M and its radius is R, what is its density?

b. Using the shell theorem, calculate the gravitational force the Earth exerts on a small particle of mass m located a distance r from the center of the Earth (r < R).

c. Use the result of part b to calculate the work required to move a particle of mass m from the center of the Earth to the surface. If you were not able to obtain the answer to part b, assume that the force in the interior of the Earth can be written as a r and adjust a to ensure that force at r = R agrees with what you know about the gravitational force on the surface.

d. Compare the work calculated in part c with the work required to move a mass m from the surface of the Earth to infinity (r ® ¥).

Express all your answers in terms of m, M, r, R, g, and G.

Problem 16 (25 points)

A block of mass M is attached to a spring with spring constant k. The spring-mass system is in equilibrium when is has a length L. A bullet of mass m is fired from below and buries itself in the block. The block reaches a maximum height h above its original position.

a. What is the rest length of the spring?

b. What is the speed of the block right after the bullet hits?

c. What is the speed of the bullet just before it hits the block?

Express all your answers in terms of k, L, h, m, M, and g.

Problem 17 (25 points)

A plank, of length L and mass M, rests on the ground and on a frictionless roller at the top of a wall of height h (see Figure). The center of gravity of the plank is at its center. The plank remains in equilibrium for any value of q ³ q ₀ but slips if q < q ₀.

a. Calculate the magnitude of the force exerted by the roller on the plank when q = q ₀.

b. Calculate the magnitude of the normal force exerted by ground on the plank when q = q ₀.

c. Calculate the magnitude of the friction force between the ground and the plank when q = q ₀.

d. Calculate the coefficient of static friction.

Express all your answers in terms of M, L, h, g, and q ₀.