Question

A low-friction cart of mass m rests on a horizontal table. The cart is attached to a relaxed light spring constant k. At distance d from the first cart rests a second identical cart. Both cars are covered with Velcro so they stick together if they collide or touch. The first cart is pushed to the left with initial speed v0. b) determine the amplitude of a vibrating system. Consider the case when the right cart does not reach the left cart.

b) Determine the amplitude of a vibrating system

Answer 1

When the right cart does not reach the left cart, it is a single mass spring system
The angular frequency of such system is, w = sqrt(k/m)
frequency, f = w/2pi
f = (1/2pi)*sqrt(k/m) = (1/2pi) √(k/m)
Initial energy of the spring mass system = 1/2 mv0^2
Final energy = 1/2kA^2 (kinetic energy becomes zero and only potential energy stored in the spring remains)
Amplitude = A
1/2 mv0^2 = 1/2kA^2
A = vo √(m/k)