A 10.6-kg object oscillates at the end of a horizontal spring that has a spring constant of 2.05 x 104 N.m. The effect of friction force is represented by the damping coefficient b= 3.00 N.s.m!. C. 1) Set up the differential equation of motion for oscillations of the system. C. 2) Show that x(t) = Aoe- equation found in part C. a). C. 3) Draw a representative figure for the damped vibration x(t). C. 4) Calculate the angular frequency wo (o 00) and the period T. -bt/2m cos(wt + @) is a solution for the previous %3D Calculate the frequency of the damped oscillation.

A spring with spring constant 18.0 N/m hangs from the ceiling. A 510 g ball is attached to the spring and allowed to come to rest. It is then pulled down 7.80 cm and released. Part A What is the time constant if the ball's amplitude has decreased to 2.60 cm after 53.0 oscillations? Express your answer with the appropriate units. HA Value Units

I would like some help with thus problem, thank you!

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.