PH 101 Tutorial 4 Date: 28/08/2017 1. A simple way to measure the speed of a bullet is with a ballistic pendulum. As illustrated in Fig. 1 this consists of a wooden block of mass M into which the bullet is shot. The block is suspended from cables of length l and the impact of the bullet causes it to swing through a maximum angle φ as shown. The initial speed of the bullet is v and its mass is m. (a) How fast is the block moving immediately after the bullet comes to rest? (Assume that this happens quickly.) (b) Show how to find the velocity of the bullet by measuring m M l and φ. Fig. 1 2. A block of mass M on a horizontal frictionless table is connected to a spring (spring constant k). The block is set in motion so that it oscillates about its equilibrium point with a certain amplitude A!. The period of motion is T! = 2π M/k (a) A lump of sticky putty of mass m is dropped onto the block. The putty sticks without bouncing. The putty hits M at the instant when the velocity of M is zero. Find (1) The new period. (2) The new amplitude. (3) The change in the mechanical energy of the system. (b) Repeat part (a) but this time assume that the sticky putty hits M at the instant when M has its maximum velocity. 3. A particle of mass m and velocity v! collides elastically with a particle of mass M initially at rest and is scattered through angle Θ in the center of mass C system. (a) Find the final velocity of m in the laboratory L system. (b) Find the fractional loss of kinetic energy of m. 4. A superball of mass m bounces back and forth with speed v between two parallel walls as shown in Fig. 2. The walls are initially separated by distance l. Gravity is neglected and the collisions are perfectly elastic. (a) Find the time average force F on each wall. (b) f one surface is slowly moved toward the other with speed V v the bounce rate will increase due to the shorter distance between collisions and because the ball s speed increases when it bounces from the moving surface. Show that!" =!"!" =! and find v(x).!"!!"! (c) Find the average force at distance x. Fig. 2 5. Two balls of mass m and mass 2m approach from perpendicular directions with identical speeds v and collide. After the collision the more massive ball moves with the same speed v but downward perpendicular to its original direction (Refer to Fig. 3). The less massive ball moves with speed U at an angle θ with respect to the horizontal. Assume that no external forces act during the collision. 1
PH 101 Tutorial 4 Date: 28/08/2017 Fig. 3 (a) Calculate the final speed U of the less massive ball and the angle θ. (b) Determine how much kinetic energy is lost or gained by the two balls during the collision. s this collision elastic inelastic or superelastic? 6. (a) A particle executes Simple Harmonic motion and is located at x = a b and c at time t! 2t!!!!!!! and 3t! respectively. Show that the frequency of oscillation is cos. [Hint: take the SHM!!!!!! to be of the form x = A sin(ωt)] (b) A cylinder of mass m is allowed to roll on a smooth horizontal table with a spring of spring constant k attached to it so that it executes SHM about the equilibrium position. Find the time period of oscillations. 7. A U tube with cross section A is filled with a liquid (having density ρ and total mass M) the total length of the liquid column being h. f the liquid on one side is slightly depressed by blowing gently down the levels of the liquid will oscillate about the equilibrium position before finally coming to rest. (a) Show that the oscillations are SHM. (b) Find the period of oscillations. 2
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