1) A crate is pulled along a horizontal surface at constant velocity by an applied force F_a that makes an angle q with the horizontal. The coefficient of kinetic friction between the crate and the surface is m. Show that the magnitude of F_a is a minimum when the angle q is given by tan^-1 m.

2) Suppose you are sitting in a car that is speeding up. Draw well-separated force diagrams of the following objects:

• your own body;
• the seat in which you are sitting (apart from the car);
• the car (apart from the seat);
• the road surface where the tires and the road interact.

Assume the car has rear wheel drive.

• Describe each force in words; show larger forces with longer arrows.
• Identify the third law pairs.
• Explain carefully in your own words how the force imparting acceleration to the car originates.

3) A block of mass M is pushed along a frictionless table by a force F for a distance s as shown in the figure at the right. The force is inclined to the horizontal at an angle q. When it reaches s, the force is removed. The block starts from rest at clock time 0:00. Express your answers in terms of M, F (= the magnitude of F), q, and s.

(a) Calculate the acceleration of the block during the time the force is acting.
(b) At what time will the box have moved through the distance s? Find the object's velocity at that time.
(c) Calculate the change in the object's kinetic energy using the work-energy theorem and show that it gives the same result you would calculate using change in the object's velocity.
(d) What is the object's change in momentum from the time it starts to the time it reaches s?
(e) If the same force pushed a larger mass for the same distance, would the object have more or less kinetic energy than the original box? What about momentum? Give a qualitative explanation for your results that makes them plausible.

4) Newton's first law states that an object will move with a constant velocity if nothing acts on it. This seems to contradict our everyday experience that all moving objects come to rest unless something acts on it to keep it going. Does our everyday experience contradict one of Newton's Laws? If it does not, explain the apparent contradiction. If it does, explain why we bother to teach Newton's first law anyway.

5) A worker is pushing a cart along the floor. At first, the worker has to push hard in order to get the cart moving. After a while, it is easier to push. Finally, the worker has to pull back on the cart in order to bring it to a stop before it hits the wall. The force exerted by the worker on the cart is purely horizontal. Take the direction the worker is going as positive.

Below are shown graphs of some of the physical variables of the problem. Match the graphs with the variables in the list below. You may use a graph more than once or not at all. (Note: the time axes are to the same scale, but the ordinates {"y axes"} are not.)

(a) friction force
(b) force exerted by the worker
(c) net force
(d) acceleration
(e) velocity.

6) You are pushing on a block that is resting on the table.

(a) You press on the block but not hard enough to start it moving. What are the forces acting on the block? (Be sure to specify the type of force and the object causing each force.) Wherever you can, compare the magnitudes of forces. Give a brief explanation for how you know each of the results you state.
(b) You press a bit harder and the block begins to move. After a moment of starting up, you press so that the block is moving with a constant velocity. For the time while the block is moving with a constant velocity have any of the comparisons you stated in (a) changed? Which ones? Have any of the forces changed? Which ones?
(c) Suppose the block has a mass of 0.4 kg and the coefficient of friction between the block and the table is 0.3. What force will you have to use to keep it going at a constant velocity of 0.2 m/s? (You may take g = 10 N/kg.)

7) A block of mass M1 is sitting on a frictionless table. It is connected by a massless string over a massless and frictionless pulley to another block of mass M2. (a) Build free-body diagrams for each of the masses and write equations of motion for each object. Use the coordinate x1 shown in the figure for the position of mass M1 and coordinate y2 shown in the figure for the position of mass M2. (b) Use these equations of motion to obtain the acceleration of the two objects. Explicitly state any conditions that you are applying to solve the equations.

8) A block of mass M₁ is sitting on a table with a block of mass M₂ sitting on top of it. Attached to the bottom block is a ring and a string is attached to the ring. The string is pulled with a tension T. The coefficient of friction between block 1 and the table is m₁, and the coefficient of friction between block 2 and block 1 is m₂. (Ignore the difference between static and kinetic friction.)

(a) For one given value of T the blocks accelerate together.

• Draw free body diagrams for each of the blocks.
• Calculate the acceleration of the blocks.
• What can you say about the magnitude and direction of the various frictional forces in the system?

(b) For a second, larger, value of T, the first block continues to accelerate, but the second block begins to slip back on the first block.

• How do the free body diagrams for the two blocks change from the case above?
• Calculate the acceleration of block 2.

9) The system shown in the figure is initially motionless and the pulleys are of negligible mass. (a) Write free-body diagrams for each mass and their equations of motion. (b) Show that the accelerations of the three objects are related by a1 + a2 + 2 a3 = 0 (c) Suppose m2 = 2 m1. When the system is released, it is found that body 1 remains stationary. What is the tension in the string supporting body 1? (d) Find the magnitude and direction of the acceleration of body 2, of pulley A, of body 3, and the tension in the string supporting pulley A. (e) What is the mass of body 3?

10) Write an essay about Newton's second law. In your essay, make sure you state the law in words and equation form. For the equation you give, explain carefully the meaning of each symbol in the equation. Also include a brief discussion of "what the equation is good for", that is, how you might use it in practice.

11) A physics student is pulling a small wooden crate across a wooden floor using a rope tied to a ring which is attached to the box.

The box has a mass m, the student has a mass M, and the coefficient of friction between the box and the floor is m. (Ignore the difference between static and kinetic friction.) The angle between the rope and the horizontal is q.

(a) What force does the student have to exert on the rope to keep the box moving at a constant speed?
(Express your answer in terms of the relevant given symbols.)
(b) If q = 30º, m = 20 kg, M = 80 kg, m = 0.4, and g = 10 N/kg, find the tension in the rope.
(c) State one approximation you have made in order to make this problem more easily solved. (By "approximation" I mean some simplification that has been made or real-world effect that is being ignored. I do not mean that one of the numbers given might be a bit wrong.)

13) Two blocks, A and B, of masses M_A and M_B respectively, are tied together with a rope, R, of mass M. The small block, B, is being pushed with a constant horizontal force as shown below. Assume that there is no friction between the blocks and the table, and that the blocks have already been moving for a while at the instant shown and their relative position is not changing. The "push" is exerting a force F on block B.

(a) Will block B be moving at a constant speed or will it be speeding up? Explain your answer.
(b) Draw careful free-body diagrams specifying all the forces acting on the three objects in the problem.
(c) Calculate the acceleration of the system.

14) A large heavy cart of mass M is sitting on a table at rest. The cart has wheels and rolls on the table with negligible friction. A smaller block of mass m is tossed so it lands on top of the cart at time t = 0 . At this instant, when the block first touches the cart, the cart is at rest and the block is moving with a velocity v0 as shown in the figure below. The coefficient of friction between the cart and the block is m.

(a) Assume that the block slides on the cart for a while before being brought to a stop relative to the cart. (Assume the block is not moving fast enough to slide off the end of the cart.) Describe what happens to the cart and the block. (b) While the block is sliding on the cart, draw separate free-body diagrams for the cart and the block, showing all forces that act on each one. (c) Is the momentum of the system consisting of the cart + block conserved? Explain. (d) Find the final velocity of the cart-block system after the block has come to rest on top of the cart. Express your answer in terms of the symbols given in the description of the problem.

15) The figure to the right shows a multiple-exposure photograph of a ball rolling up an inclined plane. (The ball is rolling in the dark, the camera lens is held open, and a brief flash occurs every 3/4 sec four times.) The left-most ball corresponds to an instant just after the ball was released. The right-most ball is at the highest point the ball reaches. (a) Copy this picture and, at each ball, draw an arrow to indicate the velocity of the ball at the instant when it was at that point in space. Explain what is happening ("tell the story" of the picture). (b) For the instant of time when the ball is at the second position shown from the left, draw a free-body diagram for the ball and indicate all forces acting on it. (c) If the mass of the ball is m, what is its acceleration? (d) If the angle q is equal to 30o, how long is the distance s?

16) Two identical billiard balls are labeled A and B. Maryland Fats places ball A at the very edge of the table, ball B at the other side. He strikes ball B with his cue so that it flies across the table and off the edge. As it passes A, it just touches ball A lightly, knocking it off. The balls are shown just at the instant they have left the table. Ball B is moving with a speed v0, ball A is essentially at rest. (a) Which ball do you think will hit the ground first? Explain your reasoning. Below are shown a number of graphs of a quantity versus time. For each of the items below, select which graph could be a plot of that quantity vs. time. If none of the graphs are possible, write N. The time axes are taken to have t=0 at the instant both balls leave the table. Use the x and y axes shown in the figure. For each of the following, which graph could represent (b) the x-component of the velocity of ball B? (c) the y-component of the velocity of ball A? (d) the y-component of the acceleration of ball A? (e) the y-component of the force on ball B? (f) the y-component of the force on ball A? (g) the x-component of the velocity of ball A? (h) the y-component of the acceleration of ball B?

A B C D E F G H

18) You and your friends have prepared a large pot of soup to take to the local homeless shelter. The pot is large two feet high and two feet in diameter. In order to get it to the shelter, you put the pot in the back of your friend's pick-up truck, pressed against the back wall (next to the cab) and against the left wall (on the driver's side). You are pretty sure that you have to drive carefully so the pot doesn't tip over. Do you have to be more careful when starting or when stopping? When turning left or when turning right? Explain your choice.