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? [a] 10 6 [d] 4 [b] 10 6 [e] 10 6 [c] 4 [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. [a] 2 [b] [c] [d] 2 [e] [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? [a] [b] [c] 0 [d] [e] [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? [a] The cylinder [b] The hollow cylinder [c] The sphere [d] The hollow sphere [e] They will all have the same speed [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: [a] The total mechanical energy is conserved; [b] The total momentum is conserved; [c] The velocity of the center of mass is conserved; [d] The total energy is conserved; [e] The total gravitational potential energy is conserved; 2
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[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. --------------------------------------------------------------------------------------------------------------------- 4
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[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. --------------------------------------------------------------------------------------------------------------------- 6
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[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. --------------------------------------------------------------------------------------------------------------------- 8
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[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. --------------------------------------------------------------------------------------------------------------------- 10
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EQUATION SHEET (Make sure to start all problems from these equations only) Vectors: Linear motion: 2D:, ; tan ; cos, sin. ; ; Relative velocity: Newton s Laws: Examples of forces: ; ; ; ; 0 0; ; ; ; friction:, 0, Work: ; ; Power: Work-Energy Theorem: ; conservative forces: in general: or ΔΔ Momentum and impulse: Center of mass: examples: ; ; Force: ; or ; Circular motion: ; ; ; ; Uniform circular motion: ; ; ; 2; Rolling without slipping: Moment of inertia:, or ; Energy of rigid bodies: ; 12