Signature: I.D. number: Name: 1 You must do the first problem which consists of five multiple choice questions. Then you must do three of the four long problems numbered 2-5. Clearly cross out the page and the numbered box of the problem omitted. Do not write in the other boxes. If you do all the problems, only problems 1-4 will be graded. Each problem is worth 25 points for a total of 100 points. TO GET CREDIT IN PROBLEMS 2 5 YOU MUST SHOW GOOD WORK. 2 3 4 CHECK DISCUSSION SECTION ATTENDED: [ ] Dr. Nepomechie 2O, 9:30 10:20 a.m. [ ] Dr. Voss 2Q, 12:30 1:20 p.m. [ ] Dr. Voss 2R, 2:00 2:50 p.m. [ ] Dr. Huffenberger 2S, 3:30 4:20 p.m. [ ] Dr. Galeazzi 4P, 11:00 11:50 a.m. 5 TOTAL THE EQUATION SHEET IS PROVIDED ON THE LAST PAGE WHICH YOU CAN TEAR OFF. 1
[1.] This problem has five multiple choice questions. Circle the best answer in each case. [1A.] A particle with mass 2, is initially moving with velocity 5. What is the impulse necessary to change the particle velocity to 3? 2 3 5 10 6 [1B.] The potential energy associate with the force is given by,,, with and constants. Find the force as a function of x, y, and z. 2 [1C.] Two friends, Alice and Ben, are initially resting together on the ice rink, when Alice decides to push Ben away. If Alice s mass is, Ben s mass is and there is no friction on the ice rink, how far has Alice moved, if any at all, after Ben has traveled a distance d? There CM does not move, placing it in the origin: 0 [1D.] A cylinder, a hollow cylinder, a sphere, and a hollow sphere, are racing down an incline with friction. Which one will be faster at the bottom? The object with the smallest moment of inertia will have the smallest rotational kinetic energy, thus the biggest linear energy (and biggest speed): that is the sphere. [1E.] A bullet is fired horizontally into a wood block and gets embedded into it. The wood is placed on a flat ice surface and is free to move without friction. Which of the following statements IS NOT true about the bullet plus block system: This is a fully inelastic collision and the total mechanical energy IS NOT conserved. 2
[2.] A hollow cylinder with mass M and radius R is initially at rest. The cylinder is spun around its axis at constant angular acceleration until it reaches an angular velocity. [a] How long does it take for the cylinder to be accelerated to its angular velocity? The spinning cylinder is then placed on a flat surface where it starts rolling until it moves without slipping. Assume that the work done by friction between the cylinder and the surface is negligible. [b] Derive the moment of inertia of the cylinder. [c] Find the kinetic energy of the cylinder before it is placed on the flat surface. [d] Find the speed of the cylinder s center of mass when it moves on the surface without slipping. Write your results in terms of M, R,, and, and MAKE SURE TO SHOW YOUR WORK. Remember to check the units/dimensions for each answer. [a] (constant) Check dimensions: [b] In a hollow cylinder all the mass is at the same distance R from the axis of the cylinder. The moment of inertia is therefore simply. Check dimensions: Moment of inertia has units of mass times distance squared, therefore units are [c] Check dimensions: [d] Conservation of energy:. When the cylinder rolls without slipping or Check dimensions: 3
[3.] Consider a rod of length L with the left end at the origin of a Cartesian coordinate system. The mass per length of the rod changes as a function of the distance from the origin according to the expression: With A positive and constant. [a] Find the mass of the rod. [b] Find the position of the center of mass of the rod. [c] Find the moment of inertia of the rod with respect to the z-axis (through the origin). [d] Find the moment of inertia of the rod with respect to an axis parallel to the z-axis and passing through the center of mass of the rod. Write your results in terms of L and A, and MAKE SURE TO SHOW YOUR WORK. Remember to check the units/dimensions for each answer. [a] Check dimensions: units of A [b] Check dimensions: [c] Check dimensions: [d] Check dimensions: 4
[4.] On a flat, frozen surface, a hockey puck (#1) with mass m moves with speed toward a second puck (#2), which is at rest and has the same mass. The collision between the pucks is head-on and can be considered elastic. Assume there is no friction. [a] Derive the velocity of puck #1 after the collision; [b] Derive the velocity of puck #2 after the collision; [c] Derive the velocity of the center of mass of the two pucks before the collision; [d] Derive the velocity of the center of mass of the two pucks after the collision. Write your results in terms of m and, and MAKE SURE TO SHOW YOUR WORK. If you exclude any solution, explain the physical reasoning for that. Remember to check the units/dimensions for each answer. 2 2 2 0 #1 0 ; #2 0 We can exclude solution #2 as it represents the case where the first puck misses the second and therefore there is no collision (the first puck keeps moving at the same velocity, the second stays at rest). Therefore: [a] 0 [b] Check dimensions: [c] Before the collision: Check dimensions: [d] After the collision: There are no external forces therefore the velocity of the center of mass does not change. Check dimensions: 5
[5.] A package of mass m is released on a ramp, a distance L from the top of a long spring with constant k attached to the bottom of the ramp. The ramp forms an angle with the horizontal plane and there is no friction between the package and the surface of the ramp. [a] Find the speed of the package just before hitting the spring. [b] Find the maximum compression of the spring (don t forget gravity!). [c] After the spring is compressed it will push the package back up the ramp. What is the speed of the package right when the spring is fully decompressed again? Write your result in terms of m, L, k,, and g, and MAKE SURE TO SHOW YOUR WORK. Remember to check the units/dimensions for each answer. [a] Conservation of energy. Placing the origin at the top of the unstretched spring: sin 2 sin Check dimensions: [b] Conservation of energy again: sin sin Where x is the compression of the spring (positive). Solving for x: The solution with the minus sign gives a negative result, therefore only the positive one is acceptable (we are calculating a positive distance), thus Check dimensions: 0 [c] Energy is conserved, and since the potential energy is the same as at point [a] (i.e., zero), the speed must also be the same: 2 sin. Note that the direction of the velocity is opposite. Check dimensions: same as [a] 6