AP Physics: Solving for Acceleration in a Cart Experiment

[Bayuki]

I've gone ahead and uploaded the paper I had filled out during class.

For this experiment, students must be able to find the acceleration of a cart using kinematics, dynamics, and energy. A cart with a mass on it (m_c=2.48kg) is connected to a mass-less string which is connected to a pulley (m_p=0.004kg). A hanging mass (0.02kg) is also connected to the pulley. I tried the kinematics route of solving for the acceleration and I got a=(2Δx)/t². Also, since our tables weren't high enough, we couldn't use the time for the entire cart to get to the pulley. Instead, we stopped the cart at 0.64m (which would be x) from the starting point. The time we got was 2.73s to get to 0.64m from the starting point. So using the equation I previously stated, I got 0.17m/s^2 to be my acceleration. Can someone check if my work is correct or if my equation is used correctly?

Aside from kinematics, we had to use dynamics and energy. We also did dynamics in class today. I made three free body diagrams (in the paper from class), with the cart having F_G down, F_N up, and F_T to the right. The hanging mass has F_G down and F_T up. Also, my teacher said that there must be a free body diagram for the pulley because it is rotating and taking acceleration. So the free body diagram for the pulley was F_T down, F_G down and F_N up from the center (not in the image uploaded), and with a torque going clockwise with an acceleration of α. So far, what I have tried doing is finding the F_T. I got Ft= M(cart)g - ma and Ft=m(hanging mass)a. So then I made both F_Ts equal to each other and got Mg - Ma = ma. Obviously I've got no clue as to what I'm doing at this point considering I ended up solving for acceleration instead of F_T. I still have no idea where the pulley plays into all of this. Can someone lead me in the right direction?

I also needed some help with finding acceleration using energy, and even though I haven't tried it yet, I don't know where to start ...

 

[berkeman]

Welcome to the PF.

Can you try scanning your page or somehow posting a cleared-up vertical image? It's pretty much unreadable for me.

As to the pulley, have you covered Moment of Inertia (MOI) yet in class? You will need that to help you with your torque & angular acceleration part of this problem.

 

[Bayuki]

Unfortunately all I could do is make it vertical. The actual paper is not with me so I can't scan it for a sharper image. Sorry.

As for moment of inertia, yes. We have gone over that earlier in the year (I want to say mid to late December?). We are now reviewing for the AP Physics 1 exam which will be in the first week of March. The pulley is a solid disk, so I=1/2MR^2. However, how would I tie that in with the other two (the cart and the hanging mass)? I recall some sort of equation that went like A(total)=√ac² + at², with ac being centripetal acceleration and at being transitional. Would this be something I would use?

Thanks a lot, berke.