Question

Figure 23.9 shows a sliding mass on a spring. Assume there is no friction.
Will this system oscillate? Explain why or why not.

Answer 1

Without friction, there will be undamped simple harmonic motion. The force of the spring is proportional to the distance from the equilibrium point. The period of oscillation will be independent of the amplitude.

Force = Mass*acceleration = -k*distance
m*a = -k*x ( It is a restoring force) The minus sign takes care of the direction.

When x is positive, then the force is in the negative direction, and when x is negative, the force is in the positive direction. It is called Hook's Law when this Linear relation of the Force vs Distance x holds true.
The spring constant is found from the graph of Force vs Distance X. The constant k is the slope of the line:
k = Change in force / Change in stretched distance

m*a - kx = 0 (acceleration is d^2x/dt^2)

The second order differential equation describing the motion is:
\[\frac{ d^2x }{ dt^2 }+\frac{ k }{ m }x = 0\]

Answer 2

\[x(t) = A*\cos(\omega *t + \phi)\]

A-Amplitude
omega - angular frequency
phi - phase angle

For practice, take the first and second derivative of x(t) above, and substitute those into the differential equation, you should get...
\[-\omega^2*x + \frac{ k }{ m }*x = 0\]

The only way for that to work is if :

\[\omega^2 = \frac{ k }{ m }\]

So you have the angular frequency , and the period of oscillation as:
\[\omega=\sqrt{\frac{ k }{ m }}~~~~and~~~T=\frac{ 2\pi }{ \omega } = 2\pi*\sqrt{\frac{ m }{ k }}\]

Independent of the amplitude! and the phase angle
Those are a result of only the Initial conditions. Where you start the oscillation, and what velocity you start the mass moving at.

Answer 3

Thank you so much! You're amazing :)

Answer 4

welcome, hope that helps out.