m M 4. A small block moving with a constant speed v collides inelastically with a block M attached to one end of a spring k. The other end of the spring is connected to a stationary wall. Ignore friction bet een the blocks and the surface. a. What is the speed of the system of two blocks after the collision? b. What is amplitude of oscillations of the system of two blocks? c. What is the period of oscillations? d. What is the total energy of the oscillating system?

1: In this question we will study the damped harmonic oscillator. Consider the following spring-mass system. kr bà Figure 1: Spring-mass system with a friction force. That is, an object of mass m is attached at the end of a spring with spring-constant k. We will also consider the effect of friction. Friction works in the opposite direction of motion, as illustrated in the figure. A) Show that the differential equation governing the damped spring-mass system is: më = -bå – kx (1) dr and i = dt where i = dt2 b) To solve the differential equation (1), we will take the following ansatz: r(t) = Ae. (2) If the above r(t) is a solution of Eq. (1) show that A has to satisfy the following equation: 1² +27d + wi = 0, (3) * and wi = k Solve the above equation to find A. m2 b where y 3= 2m c) For wi >?, show that the general solution can be written as: x(a) = Ae-t cos (wt +0) (4) where w? = w - 7². 2: Using Fermat's principle prove the law of reflection and the law of refraction of light.

A 500g block connected to a spring which the force constant is 5 N/m is free to oscillate on a frictionless, horizontal surface. The block is displaced 6cm from equilibrium and released from rest. Calculate the following: a.Period ofthe motion in seconds b.Maximum speed in m/s. c.Maximum acceleration in m/s2 d.Express the position, velocity, and acceleration as functions of time.

A mass weighing 4 Ibs. stretches a spring 6 in. If the mass is pushed upward, contracting the spring distance 1 in and then set in motion with a downward velocity of 2 fl/s, and if there is no damping. a. Find the position function u of the mass at any time t. b. Determine the frequency, period, amplitude, and phase of the motion.

Letters A-D if possible

Part a and part b