Simple Harmonic Motion & a vertical spring

[danielle36]

Homework Statement

A spring is standing upright on a table with its bottom end fastened to the table. A block is dropped from a height of 3.0 cm above the spring. The block sticks to the top end of the spring and then oscillates with an amplitude of 10 cm. What is the oscillation frequency?
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My first thought was I should be applying energy equations to solve...

$$U_{g_{i}} = K_{f}$$
$$mgy = 1/2mv^{2}$$
$$(9.81m/s^{2})(.03m) = 1/2v^{2}$$
$$v_{max} = 0.767 m/s$$

$$v_{max} = 2 \pi fA$$
$$f = 1.22 Hz$$


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The answer is supposed to be 1.83 Hz..

I'm thinking I'm running into trouble because I made the assumption that the velocity just before the block hits the spring will be the same as the max velocity for the oscillation, but this is the only way I can think to do this question since I don't know the spring constant

Any help would be appreciated

 

[alphysicist]

Hi danielle36,

That is where your mistake is. The block does not begin slowing down as soon as it hits the spring. It will not begin slowing down until the spring force upwards is greater than the gravity force downwards.

This means the equilibrium point is some distance d below where the block first touches the spring, and that's where the maximum velocity occurs. If you follow the same basic idea that you did in your post, but choose the real equilibrium point, you can get the right answer. (To solve it I think you'll also have to use some other facts about the spring, like what defines the equilibrium point and how the frequency is related to the spring constant.) What do you get?

 

[danielle36]

Well I've tried a few things and I still haven't found the right method... I don't know of a way to relate the frequency and spring constant, so I tried using (mg/k)$$^2$$ as my value for v in 1/2mv$$^2$$ but it didn't leave me with anything useful... So I'm really not sure where to go from here

 

[alphysicist]

Well I've tried a few things and I still haven't found the right method... I don't know of a way to relate the frequency and spring constant, so I tried using (mg/k)$$^2$$ as my value for v in 1/2mv$$^2$$ but it didn't leave me with anything useful... So I'm really not sure where to go from here

I would try writing the energy conservation equation at the beginning point and at the equilibrium point. You've already written down the equation relating vmax and amplitude/frequency, so you can get rid of vmax, for example, in terms of f and A.

That still leaves quite a few unknown, which you can get rid of by using other relations that you know. For example, if the equilibrium point is a distance d below where the spring starts, how is d related to other parts of the problem? Another important thing is how is the frequency related to the spring constant?

Anyways, by starting with the energy equation I think you should be able to get rid of all the unknowns except the frequency. What do you get?