# SHM with mass over pulley problem

1. Dec 6, 2004

### gaborfk

I am having a problem with this:

One end of a light spring with force constant 100N/m is attached to a vertical wall. A light string is tied to the other end of the horizontal spring. The string changes from horizontal to vertical as it passes over a solid pulley of diameter 0.04m. The pulley is free to turn on a fixed smooth axle. The vertical section of the string supports a 0.2kg mass. The string does not slip at its contact with the pulley. Find the frequency of oscillation of the object if the mass of the pulley is

a) negligible

b) 0.25kg

2. Dec 6, 2004

### CartoonKid

For the part a), you can just apply the formula $$T=2\pi\sqrt{\frac{m}{k}}$$.
For part b), you have to calculate the moment of inertia of the pulley. Then apply formula $$\tau=I\alpha$$. You also need to find out the total elongation of the spring first by assuming that the mass is being let go when the spring is at equilibrium. Since the string does not slip at its contact with the pulley, this elongation will tell you something on the pulley.

3. Dec 6, 2004

### gaborfk

On part a)

I would not have to consider that the mass is hanging, by including gravity somehow?

4. Dec 6, 2004

### Staff: Mentor

The fact that the mass is hanging will affect the equilibrium position, but not the frequency.

One way to solve this problem is to write the equation of motion of the system and compare it to the basic dynamical equation for simple harmonic motion of a mass on a spring:
$$\frac{d^2x}{dt^2} = \frac{k}{m} x$$

5. Dec 6, 2004

### CartoonKid

If you know how to derive that formula. You will know that it's not affected by gravity.