Please label each part of the question.
2 3 Problem 2: The air-track carts in Fig.2 (m1 = 100 g, m2 = 300 g) Problem 3: A cylinder of radius R, length L, and mass M is released are sliding to the right at o = 1.0m/s. The spring between them String holding carts together m, from rest on a slope inclined at angle 0. It is oriented to roll straight has a spring constant of k = 120 N/m and is compressed by down the slope. If the slope were frictionless, the cylinder would As = 4.0 cm. The carts slide past a flame that burns through the 100 8 1 300 g slide down the slope without rotating. What minimum coefficient n Mg a string holding them together. Afterward, what are the speed of static friction is needed for the cylinder to roll down without slip- and direction of each cart? 1.0 m/s ping? a) Consider the system in the reference frame moving to- a) Fig. 3 shows the free-body diagram of the cylinder. The condi- gether with the carts to the right at their initial speed v. In this tion for the cylinder to roll down without slipping is a, = a, where reference frame, the initial momentum and the initial kinetic FIG. 2: The scheme for Problem 2 at = QR is the tangential acceleration of the rim of the cylinder, a energy of the system are both zero, and its initial potential energy is the one of the spring. After the string is the cylinder's angular acceleration, and a is the acceleration with FIG. 3: The scheme for Problem 3 burns, the final momentum of the system stays zero, while the potential energy of the spring goes into which its center of mass moves down the slope. the kinetic energy of the carts (the total mechanical energy is conserved). From the conservation of mo- The only force creating the torque about the center of mass is the static friction force fs (the other mentum and energy, derive the expressions for the final velocities vix and vzx of the carts in this (moving) two forces point radially towards or away from the center of mass). Use the formula for the angular reference frame in terms of k, As, m, and m2. (Partial answer: vix = - K(As) 2 acceleration, a = /I, where r is the torque and I = - MRZ is the moment of inertia of the cylinder, to get V mi(1+ miz ) an expression for at in terms of fs and M. b) The acceleration a is related to the forces along the slope by Newton's second law. Write down Newton's second law along the slope and impose the condition a = at. Using your formula for a, from the previous step, show that for the rolling without slipping the friction force has to be fs = } Mg sin 0. b) Transform these velocities into the reference frame stationary with respect to the Earth and compute their numerical values. (Answer: -0.2m/s and 1.4 m/s.) c) Show that the minimum static friction coefficient for which the rolling without slipping is possible (the static friction force has to be at its maximum, fs = nus) is Hs = 3 tan 0
