3. [30 + 5 marks] A block of mass m oscillates on a spring with spring constant k. The surface has no friction. The oscillating motion initially has amplitude A. At t = 0, when the mass is at position +A/2 and moving to the right (positive velocity) as shown in the upper part of gure 1b. it collides with a second block of equal mass V a (t ) stick (a) Torsional har- (b) Collision of two masses (Q3) monic oscillator (Q2) Figure l: Sketches for questions 2 and 3. m. The blocks stick together to form a new block of mass 2m and continue to the right as shown in the lower part of gure lb. We will denote quantities before the collision with a subscript \"1\" and quantities after with subscript \"2". We assume that momentum is conserved in the collision, i.e., mm = vag and we have 122 : 121/2 the instant after the collision. (a) [5] Let wl : \\/ k/ m be the oscillation frequency before the collision. What is the new oscillation frequency (1)2 after the collision? (b) [13] Suppose the motion before the collision is given by x1 (t) = Acos(w1 t + (/2). Find (/2, and then an expression for 121 (t = 0), followed by 1120' = 0) the instant after the collision. Except for (,b, express all results in terms of ml and A. Hint: cos(7t/3) = 1/2 and sin(n/3) = \\//2. (c) [12] Find an expression for x2(t), the subsequent motion of the joined masses after the collision. Hint: Since the collision is at t = 0, velocity U2 and the collision position x = +A/ 2 constitute initial conditions on the subsequent motion. You may want to use the general form x(t) = B cos(w t) + C sin(a) t), where B and C are constants. (d) [5 bonus marks] What is the new amplitude of the subsequent undamped mo- tion, expressed in terms of A? Hint: Remember that the new amplitude is just the maximum of x2 (I)