Problem 1: In Fig. 1, nd an expression for the acceleration of mi. The pulleys are massless and frictionless. a) Write down the relation between the magnitudes of the ac- celerations of the two blocks, a, and a; (it is not :21 : a2, and the I": ******* Vectors in Fig. 1 are not drawn to scale). An argument that could help is that the total length of the rope stays constant during the motion. Friflillnlk'\\.\\ b) Write down Newton's second law for each block. Do not miss 515- 1: The \"heme f0! Pmblem 1 the fact that block m2 experiences tension forces from both ends of the rope passing through its pulley. Using the acceleration constraint from part a), work out the formula for the acceleration a, in terms of m1. m2, and g. c) What is the value on\Problem 2: Two wires are tied to the sphere with mass m shown in Fig. 2. The sphere revolves in a horizontal circle at a constant speed 11. What is the tension in each of the wires? a) The forces acting on the sphere are shown in Fig. 2 (notice that \"gm we expect tension 7'} to have a smaller magnitude than tension :71). Write down r and z-components of N ewton's second law for this mass (the only acceleration in this system is centripetal, in positive r-direction). You should get a system of two equations, where the two unknown quantities are the magnitudes T1 and T2. while all the other parameters are either given in the description of the problem or dened by the geometry (m, u, R, g, and the angle 0). (15m FIG. 2: The scheme for Problem 2 b) Solve this system of equations to get the formulae for T1 and T2 in terms of other parameters. (Partial . _ n v'_ _9_ answer. T, 7 2 (Moss + m9).) c) Compute the values of T, and T2 for m : 0.5 kg and u : 7.8m/s. The numerical values for R and 6 can be derived from the scheme. (Answer: T, = 25 N, T2 = 15 N.) Problem 3: Fig. 3 shows a small block of mass m sliding around the inside of an L-shaped track of radius R. The bottom of the track is frictionless; the coe'icient of kinetic friction between the block and the wall of the track is pk. The block's speed is on at t = 0. Find an expression for the block's speed, of, at a later time If. w Bolluni n / t'nmunlcss \"'E a) The forces acting on the block are shown in Fig. 3. The force \"6' 3: The \"heme for Pmblem 3 I'Iw is the normal surface force exerted on the block by the wall. and it is the only force responsible for the centripetal acceleration of the block. The force f; is the kinetic friction between the wall and the block, and it is the only force responsible for the tangential acceleration of the block, a, = dv/dt = (/m, where u is the speed of the block at any given moment of time (the minus sign in the last equality comes from the fact that the tangential component of f; is negative, and therefore the speed of the block is decreasing with time, not increasing). With this information show that the speed of the block obeys the following differential equation: :10 v2 dr''ki' b) Multiplying this equation by dt, dividing by 02, and integrating both sides from the initial state of the system (time t = 0, velocity on) to its nal state (time t = tf, velocity vf) we get fw fled. \"a 02 a R ' Perform the integration and show that the expression for the nal speed becomes 0/ : "\"R anitkll