Classical mechanics - mass/spring attached to moving support

[ihatelolcats]

Homework Statement

A mass, m, is attached to a support by a spring with spring constant, k. The mass is hanging down from the spring, so there is a gravitational force on the mass as well. Neglect any resistive or frictional force. The support is then oscillated with an amplitude of A and at a frequency of ωa .
a) Find the motion of the mass relative to the support.
b) Find the motion of the mass relative to the lab (inertial frame).

Homework Equations

F=ma

The Attempt at a Solution

I found the motion of the mass relative to the support to be
x*=-kAcos(ωt)/(mt^2)-(gt^2)/2
by integrating twice
ma*=kAcos(ωt)-mg
which seems reasonable.

For part b, I said
x=x*+Acos(ωt)
so the motion of the mass relative to the lab
x=Acos(ωt)(1-k/(mt^2))-(gt^2)/2.
Is this valid?

 

[BruceW]

You said that the support oscillates at a frequency of wa, (omega times a). What is a? Was it just a miss-type?

Also, for part a, I don't think you integrated cosine correctly. why does t^2 end up on the denominator of the integrated function?

 

[ihatelolcats]

You are correct, the frequency is just w. And it should be ω^2 instead of t^2 in the denominator.
So the answer to part b is changed to
x=Acos(ωt)(1-k/(mω^2))-(gt^2)/2.

 

[BruceW]

Unfortunately, its not so simple. This equation: ma*=kAcos(ωt)-mg is wrong. It assumes that the only contribution to the relative acceleration is due to the support moving, but the mass will have motion also.

Sorry I didn't say this in my first post, but I had only looked through your work for obvious errors. Another reason to be suspicious of your answer is that as t gets larger, x approaches minus infinity! This is not a realistic solution, so it should hint to you that your assumptions were incorrect.

The true equation representing this situation is an inhomogeneous 2nd-order differential equation. To get the equation, just think about the position of the support and position of the mass as two separate variables (to begin with), and you know the equation for the force on the mass due to relative distance between mass and support. Then you can see the position of the mass as the dependent variable and the position of the support as simply the inhomogeneous part of the equation. From there you need to solve the complementary function and particular integral, which eventually gives you the solution.

There are probably short-cut ways of doing all this, but this is the standard way that is shown in most textbooks.