Q1 A spring having a spring constant of 250 is used in a simple set of scales to measure the weights of objects placed on the pan. The pan (of mass 0.5 kg) rests on top of the spring as illustrated.
A block of mass M kg is placed on the pan, causing the spring to oscillate. There is a damping force of 10v Newton, where v m/s is the speed of the pan.
a) Determine the differential equation of motion for the damped oscillations;
b) Show that the general solution of the equation found in item (a) has the form x = Aoe-10t/(2m+1)cos(ωt + δ), where Ao is the initial amplitude. Find an expression for w in terms of M; (4')
c) Find the time (in term of M) it takes for the system to settle down to oscillations of only 25% of the initial amplitude; (4')
d) Interpret the effect if remove the damping force have on the system?