Block on a vertical spring, finding frequency

[laurenm02]

Homework Statement

A block of mass m is on a platform of mass M, supported by a vertical, massless spring with spring constant k.

When the system is at rest, how much is the spring compressed?
When the spring is pushed an extra distance x, what is the frequency of vertical oscillation?

Homework Equations

F = -kx

The Attempt at a Solution

For the first question, I set F = mg = -kx, where m = (M + m), and found x = (M+m)g/-k

For the second part, I'm not sure how to set it up. I first thought that frequency is independent of amplitude, but I was told that I have to first write a new second law equation and then find an equation of motion.

 

[sk1105]

Frequency will not be independent of amplitude, because the frequency depends on the force exerted by the spring, which in turn depends on the amplitude of the compression. So yes, you'll have to write out a new second law equation taking into account the extra compression.

 

[laurenm02]

So yes, you'll have to write out a new second law equation taking into account the extra compression

Can I get a little guidance on this? I'm not sure how to set it up.

 

[I like Serena]

Hi laurenm02! :)

Frequency will not be independent of amplitude, because the frequency depends on the force exerted by the spring, which in turn depends on the amplitude of the compression. So yes, you'll have to write out a new second law equation taking into account the extra compression.

Sorry, but frequency is independent of amplitude.

Homework Statement

A block of mass m is on a platform of mass M, supported by a vertical, massless spring with spring constant k.

When the system is at rest, how much is the spring compressed?
When the spring is pushed an extra distance x, what is the frequency of vertical oscillation?

Homework Equations

F = -kx

The Attempt at a Solution

For the first question, I set F = mg = -kx, where m = (M + m), and found x = (M+m)g/-k

For the second part, I'm not sure how to set it up. I first thought that frequency is independent of amplitude, but I was told that I have to first write a new second law equation and then find an equation of motion.

The mass M should not be included. It does not matter whether the spring rests on a platform of mass M or directly on the ground.
What matters, is that we have a mass m that is supported by the spring.
So the compression at rest should be ##\frac{mg}{k}##.

Now suppose we compress the spring by an additional amount ##x##.
Then the resultant force becomes ##F_{result} = k(x_0 - x) - mg##.
Newton's second law states that ##F_{result} = ma##.
So we get that:
$$ma = k(x_0 - x) - mg$$
Or, written as a differential equation:
$$m\frac{d^2x}{dt^2} = k(x_0 - x) - mg$$

Are you supposed to be able to solve such an equation?