Question

An object with mass 0.150 kg is acted on by an elastic restoring force with force constant 11.0 N/m . The object is set into oscillation with an initial potential energy of 0.130 J and an initial kinetic energy of 5.10×10⁻² J .

1. What is the amplitude of oscillation?

2. What is the potential energy when the displacement is one-half the amplitude?

3. At what displacement are the kinetic and potential energies equal?

4. What is the value of the phase angle ϕ if the initial velocity is positive and the initial displacement is negative?

Answer 1

The amplitude is A= max velocity/angular velocity.
So first we find max velocity. Velocity is max when kinetic energy is max.
Total max kinetic energy is
0.13+ 0.0510= 0.181 j

So put it in kinetic energy equation and find v
Kinetic energy=1/2 x m x v^2

0.181 = 0.5x 0.150 x v^2
V= 1.553 m/s.

Now lets find angular velocity w
w= (2 π )/T.
T is the time period . To find T use the formula
T= 2π under root ( m /k).
K is the force constant

This gives T= 0.733 s

So now we calculate w
w= (2 π )/0.733

So w = 8.56 rad s^-1

So as we started v =wA

1.553=8.56 x A

A= 0.181m

So we have our amplitude.


b) now the potential energy

The displacement one half the amplitude . The amplitude is is 0.181m.
0.181/2= 0.0905m
Now lets calculate the velocity at this displacement.

V= w underroot(A^2 - X^2).
X is the displacement and A is the amplitude and w is the angular velocity we calculated earlier.

V= 8.56 underoot ( 0.181^2. - 0.0905^2)

This gives
V= 0.210m/s

Calculate the kinetic energy at this velocity using

Kin energy= 1/2 m v^2

= 0.5x .250x .210^2
5.51*10^-3 j
Which gives kinetic energy
5.51*10^-3 j
As the total energy at any point is
0.181 j .
0.181- kinetic energy= potential energy

0.181-5.51*10^-3= .1755 j
Now we have our potential energy

C)
for KE and PE to be equal
they would be half the total Energy
As total energy is 0.181j
Half is 0.0905j.
Find the velocity at this kinetic energy
That is

0.0905= 1/2 m v^2

Thus gives v=1.098 m/s

The displacement x at this point is give by
V= w underroot(A^2 - X^2).

So x = 0.128m

d) the question doesn't make sense here. How can the initial velocity and initial displacement be in opposite direction in simple harmonic motion.