Problem 1 (2.5. points)

Two objects collide inelastically. Can all the initial kinetic energy in the collision be converted to other forms of energy?

1. Yes, but only for certain special initial speeds.

2. Yes, provided the objects are soft enough.

3. No, this violates a fundamental law of physics.

4. None of the above.

Problem 2 (2.5 points)

Two masses m₁ and m₂ sit on a table connected by a rope. A second rope is attached to the opposite side of m₂. Both masses are pulled along the table with the tension in the second rope equal to T₂. Let T₁ denote the tension in the first rope connecting the two masses. Which of the following statements is true?

1. T₁ = T₂

2. T₁ > T₂

3. T₁ < T₂

4. We need to know the relative values of m₁ and m₂ to answer this question.

Problem 3 (2.5 points)

A stone is launched upward into the air. In addition to the force of gravity, the stone is subject to a frictional force due to air resistance. The time the stone takes to reach the top of its flight path is

1. larger than

2. equal to

3. smaller than

the time it takes to return from the top to its original position.

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)

Some of the properties of the diatomic molecule NO can be described in terms of a simple model in which the two atoms are connected by a spring. Experiments show that the spacing between allowed vibrational energy levels in NO is 0.23 eV. Suppose you have a gas of 4 NO molecules. One molecule's vibrational energy is in the ground state (n = 0), the second molecule is in the first excited state (n = 1), the third molecule is in the second excited state (n = 2), and the fourth molecule is in the third excited state (n = 3). If we assume that all transitions from one excited states to lower-lying excited states occur with equal probability, what is the ratio of the intensities of 0.23 eV, 0.46 eV, and 0.69 eV photons?

1. 1:1:1.

2. 3:2:1.

3. 6:3:1.

4. 9:3:1.

Problem 6 (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 7 (2.5 points)

Two carts are put back-to-back on a track. Cart A has a spring-loaded piston; cart B, which has twice the inertial mass of cart A, is entirely passive. When the piston is released, cart A pushes against cart B, and the carts move apart. How do the magnitudes of the final momenta p and kinetic energies K compare?

1. p_A > p_B, K_A > K_B.

2. p_A > p_B, K_A = K_B.

3. p_A > p_B, K_A < K_B.

4. p_A = p_B, K_A > K_B.

5. p_A = p_B, K_A = K_B.

6. p_A = p_B, K_A < K_B.

7. p_A < p_B, K_A > K_B.

8. p_A < p_B, K_A = K_B.

9. p_A < p_B, K_A < K_B.

Problem 8 (2.5 points)

Two marbles, one twice as heavy as the other, are dropped to the ground from the roof of a building. Just before hitting the ground, the heavier marble has

1. as much kinetic energy as the lighter one.

2. twice as much kinetic energy as the lighter one.

3. half as much kinetic energy as the lighter one.

4. four times as much kinetic energy as the lighter one.

5. impossible to determine

Problem 9 (2.5 points)

A small rubber ball is put on top of a volleyball, and the combination is dropped from a certain height. Compared to the speed it has just before the volleyball hits the ground, the speed with which the rubber ball rebounds is

1. the same.

2. twice as large.

3. three times as large.

4. four times as large.

5. none of the above

Problem 10 (2.5 points)

I have marked my student ID number in the appropriate boxes on the scantron form, and I like to get credit for my answers on the multiple-choice questions.

1. Yes, please give me credit for these questions.

2. No, I do not need credit for these questions.

Problem 11 (25 points)

Consider and object consisting of two masses M connected by a low-mass spring of spring constant k_s. When you exert an upward force of 2Mg, the object remains at rest, as shown in the Figure below. In this situation, the spring is stretched by a distance s_i from its rest length.

At one point, a larger constant force is applied (F > 2Mg) and the object starts moving up. At some later time, the stretch of the spring has increased to s_f, and the object is located as shown in the Figure.

a. At this time, what is the speed of the center of mass of the object?

Answer:

The initial position of the center of mass of the system is located at

The initial energy of the center of mass is its gravitational potential energy, which is equal to

The final position of the center of mass of the system is located at

The final energy of the center of mass is the sum of its gravitational potential energy and its kinetic energy, which is equal to

The work done by the force F on the center of mass of the object is equal to

and is equal to the change in the energy of the center of mass:

or

The final kinetic energy of the center of mass is thus equal to

and the velocity of the center of mass is equal to

b. At this time, what is the vibrational kinetic energy of the object?

Answer:

The initial energy of the system is the sum of its gravitational potential energy and the potential energy associated with the spring. This energy is equal to

The final energy of the system is the sum of its gravitational potential energy, the potential energy of the spring, the kinetic energy of the center of mass, and the vibrational kinetic energy:

The total work done by the force F on the system is equal to

and is equal to the change in the energy of the system:

or

The vibrational kinetic energy can now be determined:

c. At this time, what is the acceleration of each block (they do not need to be the same)?

Answer:

The net force on the bottom mass is the vector sum of its weight and the spring force:

The acceleration of the bottom mass is thus equal to

The net force on the top mass is the vector sum of its weight, the spring force, and the applied force:

The acceleration of the top mass is thus equal to

Express your answers in terms of the variables provided (F, M, y_1i, y_2i, y_1f, y_2f, k_s, s_i and s_f) and the gravitational acceleration g.

Problem 12 (25 points)

Two blocks with masses m₁ and m₂, shown in the figure below, are free to move. The coefficient of static friction between the blocks is m_s but the surface beneath m₂ is frictionless.

a. What is the minimum force F required to hold m₁ against m₂?

Answer:

When we apply the external for F the system will accelerate with an acceleration a where a is equal to

The net force on block 1 in the horizontal direction must thus be equal to

In order for block 1 not to slide down, the friction force must have the same magnitude as the weight of block 1. Thus

or

The normal force and the applied force are the only two forces that act on block 1 in the horizontal direction and we must thus require that

or

The minimum force F must therefore be equal to

b. What is the acceleration of block m₂ when the minimum force F is applied to block m₁?

Answer:

When we apply the external for F both blocks will accelerate with same acceleration a where a is equal to

Express all your answers in terms of the variables provided (m₁, m₂, m_s) and the gravitational acceleration g.

Problem 13 (25 points)

Two nuclei with identical mass (M₁ = M₂ = M) are on a collision course, as indicated in the Figure below. Their initial velocities are v₁ and v₂. After the collision, they stick together.

a. What is the velocity of the center of mass of the system before the collision? Specify the magnitude and the direction of the velocity, or specify its components along the x and y axes.

Answer:

The velocity of the center of mass of the system before the collision is most easily calculated in terms of its components along the x and the y axes:

b. What is velocity of the center of mass of the system after the collision? Specify the magnitude and the direction of the velocity, or specify its components along the x and y axes.

Answer:

Since there are no external forces acting on the system, the motion of the center of mass is the same before and after the collision:

c. What fraction of the initial kinetic energy is lost in this collision? Assume that there is no rotational motion around the center of mass after the collision.

Answer:

The initial kinetic energy of the system is

The kinetic energy of the system after the collision is equal to the kinetic energy of the center of mass of the system:

The loss of kinetic energy is thus equal to

The fraction of the initial kinetic energy that is lost in the collision is equal to

Express all your answers in terms of M, v₁, and v₂.