Question

Periodic Motion

A block of mass M is attached to a horizontal spring with force constant k. It is moving with simple harmonic motion of amplitude A.

1. Calculate how much of the energy of the motion is kinetic at x= ¼ A.

2. If one adds a mass smoothly in a vertical drop at x=A, calculate what happens to A, T, and w.

Answer 1

Given system is Simple harmonic motion example

horizontal spring , Mass system making oscillations horizontally

we know that the amplitude is the maximum displacement from mean fposition
(a)

and by enerrgy conservation the total energy of the system is constant and equal to sum of k.e and Elastic potemtial energy

   E = k.e+E.p.e
and the total energy of the system at x= A is the Elastic potential energy = 0.5*k*x^2 = 0.5*k.A^2
and at mean position the total energy will be in the form of kinetic energy, E = 0.5*m*v^2

between these points the energy is sum of E.p.e and k.e

so the energy at (1/4) of the amplitude the kinetic energy k.e = 0.5*k(A/4)^2 = 0.5*k*A^2/16 = (1/16) (k.e at mean position)
or (1/16) of maximum k.e

(b)

if mass is added at position A then
A will decreases because more mass added (more inertia) so A decreases

from the relation for time period is T= 2pi sqrt(m/k)
if mass added the time period increases,

and from T and W , T = 2pi/W ==> W = 2pi/T ==> W decreases