4. We can also have damped oscillations. Consider a horizontal mass-spring system in which there is significant friction proportional to the velocity of the mass: f = -bvx where b is a constant. This force provides the damping, or a loss of mechanical energy to thermal energy. a) Using Newton's second law find a differential equation for the position of the mass. b) To solve the differential equation, try a solution with this form: x = AeBt sin (Ct + po) where A, B, C, and do are all constants. You may assume the damping is small, which means <. What must the constants B and C be such that this is a solution? c) Plot the position as a function of time. What happens to the amplitude of the oscillations as time goes on? We won't go into more detail here, but if the damping is small (as in the case above) we call the system underdamped. However, we can also have overdamped and critically damped systems, in which damping is so large (²2) the system exponentially decays instead of oscillating.

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 harmonic oscillator is made by using a 0.640 kg frictionless block and an ideal spring of unknown force constant. The oscillator is found to have a period of 0.152 s and a maximum speed of 2 m/s a)Find the force constant of the spring. Express your answer in newtons per meter.   b)Find the amplitude of the oscillation. Express your answer in millimeters.

A 2 kg mass is attached to a spring having a spring constant 8 N/m. The mass is started in motion from the equilibrium position with an initial velocity 1 m/s in the upward direction and with an applied external force F(t) = sin t. Find an expression for the displacement, x, of the mass from the equilibrium position if the force due to air resistance is - 10x N Plot x against time and interpret the main features of the graph.