Question

A child's toy consists of a spherical object of mass 50 g attached to a spring. one end of the spring is fixed to the side of the baby's crib so that when the baby pulls on the toy and lets go, the object oscillates horizontally with a simple harmonic motion. the amplitude of the oscillation is 6 cm and the maximum velocity achieved by the toy is 3.2 m/s . what is the kinetic energy k of the toy when the spring is compressed 4.5 cm from its equilibrium position?

Answer 1

Refer to the diagram shown below.

m = 50 g = 0.05 kg, the mass of the ball
A = 6 cm = 0.06 m, the amplitude

The oscillatory motion relative to the equilibrium position is
x(t) = A cos(ωt)
where 
x =  displacement at time t
A = amplitude, 0.06 m
ω = circular frequency

Therefore
x(t) = A cos(ωt) 
The velocity function is
v(t) = - ωA sin(ωt)

The maximum velocity occurs when sin(ωt) = 1. Because the maximum velocity is 3.2 m/s, therefore
(ω 1/s)*(0.06 m) = (3.2 m/s)
w = 53.33 1/s

Therefore
x(t) = 0.06 cos(53.33t)
v(t) = 3.2 sin(53.33t)

When the spring is compressed by 4.5 cm (0.045 m) from the equilibrium position, then
0.06 cos(53.33t) = -0.045
cos(53.33t) = -0.75
53.33t = cos⁻¹ (-0.75) = 2.4189
t = 0.0454 s

The velocity when t = 0.0454 s is
v = 3.2 sin(53.33*0.0454) = 2.111 m/s

The kinetic energy is
KE = (1/2)*m*v²
     = 0.5*(0.05 kg)*(2.111 m/s)²
     = 0.1114 J

Answer:  0.1114 J