Setting Up an Experiment: Force vs. Displacement

[a1234]

Homework Statement

I'm trying to figure out the setup of an experiment I have to complete. The experiment calls for a rubber band to be secured by rods on the sides, and for a clothespin to be attached to the rubber band. I have to figure out how far back the clothespin needs to be pulled down so that it just touches the ceiling once it is let go. I have to make a force vs. displacement graph to find the work done on the rubber band and set it equal to the PE of the clothespin.

Homework Equations

I'm told that Fd (on rubber band) = GPE (clothespin).
W = Fd

The Attempt at a Solution

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Is the mass in this case the mass of the clothespin? I'm thinking that the height should be the distance from the ground to the ceiling.

Fd = 0.01 kg * 10 m/s^2 * 2.5 m
Fd = 0.25

Is my setup correct?

 

[haruspex]

I'm told that Fd (on rubber band) = GPE (clothespin).

Fd is only valid if the force is constant over the distance. When you stretch a string the force increases with the stretch. The general form is ∫F.ds.
For a spring or string for which the force is proportional to the extension the integral is easy. Strictly speaking, rubber bands do not fit that model very well, but I think you are supposed to assume it does. Do you know a formula for the elastic potential energy of an ideal spring?

What are you using for d?

Another difficulty is that the rubber band has mass, so this will retain some KE after the clothespin has left. Is it light compared with the mass of the clothespin?

the distance from the ground to the ceiling.

You are asked how far you need to pull it down. That might not be all the way to the ground.

 

[a1234]

The formula for the elastic potential energy is 1/2kx^2.
I'm trying to find the value of d that would allow the clothespin to touch the ceiling upon release, so I haven't chosen a value for that.
The distance to the ceiling was chosen as the height because that is how far the pin would be from the ground once it hits the ceiling.
Is there a way other than integration to do this? I doubt we'd be expected to use integration, because this is a first-year physics course.

 

[haruspex]

The formula for the elastic potential energy is 1/2kx^2.

Right, not Fd. But what is x in that formula? Be precise.

I'm trying to find the value of d that would allow the clothespin to touch the ceiling upon release, so I haven't chosen a value for that.

I understand that you are trying to find how far down to pull the pin. Is that what you are calling d? What is the relationship between that d and the x in the formula above?

The distance to the ceiling was chosen as the height

Yes, but you wrote from ground to ceiling.
What is the set-up: are the ends of the elastic band at a fixed height and you are varying d (so not necessarily from ground), or is it always from ground but the height of the elastic band is varied (i.e. d)?

Is there a way other than integration to do this?

Assuming the elastic band behaves as a simple spring, integration produces the ½kx² formula, so just use that.

 

[a1234]

In 1/2kx^2, k refers to the spring constant and x to the amount of stretch in meters.
In Fd, d refers to the distance I need to pull back the clothespin, and x functions the same way in the above equation.
The rubber band should be placed horizontally (perhaps right next to a table with rods supporting it) so that its ends are at the same height throughout. The body of the rubber band (or clothespin) is pulled down, which changes the d value.

 

[haruspex]

In Fd, d refers to the distance I need to pull back the clothespin,

But the force you apply in pulling increases as you pull, so as I already pointed out you cannot use Fd.

its ends are at the same height throughout

Ok, so the clothespin does not necessarily go from ground. If the ceiling is height h above the initial height of the band, how high does the clothespin need to travel when it is released?

In 1/2kx^2, k refers to the spring constant and x to the amount of stretch

Right. So what is the relationship between x and d? Use geometry. You will need a variable for the initial length of the band.

 

[CWatters]

Haruspex is leading you in the right direction.

Perhaps have a think about how the force required to stretch the band changes as it's stretched. Clearly using the initial force (zero?) or the final force (maximum) wouldn't be right. If not integration what other maths function might give you a more representative figure for the force?