Coefficient of Friction of a KE-Work-PE situation

[robertsun]

Homework Statement

I have an almost-frictionless air-hockey style ramp and i place it horizontally on a table. A little cart goes on top and on one end of the cart, i attach a string which connects to a pulley and then connected to a weight, thus pulling the cart to one side with the force of gravity. The other side is connected to a spring that is attached to the end of the ramp. I put a motion detector on the end with the spring attached and i can record the motion of the cart. The experiment starts with me pulling the cart back toward the detector until the spring is fully unstretched and the detector reads zero. The cart moves back and forth in a sinusoidal pattern, but the motion dies down because of friction. I have my graph with me with values. I also have the velocity graph. This is all in logger pro. I can upload that if its needed.

Sorry that was really long. I've been going about this lab for ages. I can't seem to figure out how to do it. The point is to find the coefficient of friction.

Heres some extra data that is def needed:
mass of cart: 168g
mass of weight: 150g
location where the force of spring = force of gravity of the weight (IE the middle): 51cm
And that is where the max speed is too i think.

Homework Equations

Fspring = -kx
PE Spring= -0.5 k x^2 (also work Spring)
PE gravity = mgh
KE = 0.5 m v^2

The Attempt at a Solution

I've calculated the K to be 2.94 N/m
by -1.5N = -k x 0.51m
because in the middle, the force of the weight (1.5N) = the force of the spring (-k x 0.51)

Also, since gravity and spring are conservative, and friction is not...ill need to calculate total distance that my cart has moved...
That is the definite integral of the square root of (1 + (dx/dt)^2) with respect to t.


Thanks!

 

[SammyS]

Hello robertsun. Welcome to PF.

By what percentage does the amplitude of the oscillation decrease for each oscillation ?

There are many potential sources of friction in such an experiment. Can you be sure they're small compared to the friction between the cart & air ramp?

 

[robertsun]

our teacher just basically told us to find μk...μs being negligible and everything else is negligible. The pos decrease is quite a lot actually. Basically max at beginning is like about 1 m, and then next one is 0.9 m then the next 0.8.. Not exactly..but close

 

[ehild]

Knowing the spring constant, you can calculate the elastic energy at maximum displacement from the position where the spring is unstretched. Determine also the potential energy of the whole system (both masses) with respect to the same position. Calculate the total energy at each maximum displacements. Without friction and other dissipative forces the energy would stay constant. The difference is the work of frictional forces. Plot the energy in terms of L, the distance travelled. It should be nearly a straight line ΔE=-F(friction)*L.

ehild

 

[robertsun]

One of the problems I've encountered is how to plot the total length?

Oh, and say I want to calculate my total pot energy at the start , there is not friction, and the total energy is equal to my gravitational PE from my weight...but i don't know the height. And i can't use the next max distance traveled because a lot has turned into friction already?

Or , should i find a graph / function of the system where there is no friction. And then compare?

 

[ehild]

I hope I understand the arrangement. Was it like in the picture?

The absolute heights are not needed, as only the change of potential energy counts. You start when the spring is unstretched, and the cart is in rest. Consider that position of both the cart and the hanging weight as reference points to their own potential energy. When you release the cart, it will move uphill by distance x₁, and the hanging weight will descend by the same x₁ and the spring will be stretched also by x₁, till they come to rest and then start to move backwards. Find the potential energy both of the cart and weight with respect to their starting positions. Also find the elastic energy of the spring.
The cart moves downhill, and stops at point x₂.
Then the cart moves again upward and reaches the position x₃, and so on.

To get the distance follow the motion of the cart. First it moved from x=0 to x₁. The distance traveleld is L(1)=X₁ Then it moved from x₁ to x₂ backwards. x₂<x₁. The displacement is x₂-x₁. The distance traveled in the second step is |x₂-x₁| =x₁-x₂and the total distance traveled during the first and second steps is L(2)=x₁+x₁-x₂=2x₁-x<sub>2.
You can continue for all steps, tracing the distance traveled and calculating energy at every points xi.

ehild</sub>

 

[robertsun]

wow thanks! That helped a lot.

The only thing is (sorry i didn't make it clear) but my "ramp' is horizontal. So it just moves back and forth.

 

[ehild]

wow thanks! That helped a lot.

The only thing is (sorry i didn't make it clear) but my "ramp' is horizontal. So it just moves back and forth.

Well, I did not know what an air-hockey style ramp looks like. But why did you say that the cart starts at the top if the ramp is horizontal? That is why you should attach a figure to such questions.
If the cart moves only horizontally, everything is much easier.

ehild

 

[robertsun]

sorry for that. Should have attached a picture. As you can see I am a newbie.
But your diagram is correct...except its flat. I can use the same method correct?

 

[ehild]

sorry for that. Should have attached a picture. As you can see I am a newbie.
But your diagram is correct...except its flat. I can use the same method correct?

You can, but omit the potential energy of the cart. It does not change.


ehild

 

[robertsun]

So what I am confused about now is that if my cart moves all the way so that the spring is fully stretched..is all pot energy in the spring? Or should i calculate the weight in there somehow.

 

[ehild]

There are the potential energy of the hanging weight and the elastic potential energy of the spring.

ehild

 

[robertsun]

Alright, solved it! Thanks again

 

[ehild]

You are welcome.

ehild