Solving for g and COF on incline plane

[jasimp]

Homework Statement

The assignment is for a lab, we were to take an incline plane, and push a cart up the plane and measure the acceleration (using software). We need to use the difference in the accelerations (when the cart is moving up the plane and then back down) to find of the force of friction. We first had to develop two equations for the accelerations, up and down. I have them as a_xd=g(sin(θ)-μcos(θ)) and up, a_xu= g(sin(θ)+μcos(θ)).

I think those are right, I just changed the sign because when the cart is going up the track the force of gravity should be working with the force of friction to increase the total acceleration pulling the cart back down. However if those aren't right please let me know!

My main problem however has come from trying to solve both equations for the variables g and μ. The problem is we have to combine them and solve so that only the measured accelerations, θ and one of the variables (either g or μ are present). Once doing that I should end up with two equations that independently allow me to solve for either g or μ

Homework Equations

The Attempt at a Solution

After spending a lot of time trying to combine and solve the two acceleration equations I got

g = (a_xd+a_xu) / (2sin(θ))
μ = tan(θ)((a_xu-a_xd/(a_xd)+a_xu)

I'm not even sure if I did completely legal math when trying to solve the two equations. The reason I'm having so much trouble could be that my acceleration equations are wrong, but they seem to check out with all the research I did. Any insight into my problem would be greatly appreciated, thank you.

 

[grzz]

g = (a_xd+a_xu) / (2sin(θ))
μ = tan(θ)((a_xu-a_xd/(a_xd)+a_xu)

Your Physics and your Math looks OK.
But I am not sure whether the way you used the brackets in your tan equation is what you meant to write. I would have written it as follows:


μ = tan(θ)(a_xu-a_xd)/(a_xd+a_xu)

 

[jasimp]

Thank you for your reply. I did mess up on my brackets, thank you for pointing that out.

I just realized I left out part of my problem When I used my measured acceleration values (taken from the slope of a LoggerPro velocity graph) and then plug them into my equation solved for g I get a gravity of about 5.33.

My average measured a_xu is .2180 and my average measured a_xd is .1695. The angle of the incline above the horizon is 2.1 degrees.

 

[grzz]

May I ask how did you measure the angle of the inclined plane?

 

[jasimp]

Had the length of the hypotenuse, 122mm (a track with a mm measure along the whole thing) and then measured the height the track was raised, 4.5mm and used arcsin(4.5/122) to get 2.1 degrees.

 

[grzz]

That is a good method for the angle.
Experiments with friction are not so accurate.

 

[jasimp]

So the math looks good, and the angle is good, so it is just error in the experiment leading to such a terrible calculated gravity?

Thank you for looking this over for me.

 

[grzz]

I may have made a mistake myself. But I think that your math and physics was ok.