Calculate minimum value of force for a cart

[Inertialforce]

Homework Statement

Given the diagram shown, calculate the minimum value of h such that when the cart reaches point A, it is just about to fall off.

R = 10m
mass of cart = 158kg
note: there is no friction in the entire system

Homework Equations

ΣFc = mac
(note: Fc = centripetal force and ac = centripetal acceleration)

The Attempt at a Solution

I am not sure if I got this question right but here is what I did:

First I drew a free body diagram for the part of the diagram that had centripetal forces acting on it (included in the attachments).

Then for my calculations I started out with the equation ΣFc = mac first and here is what I did:

ΣFc = mac
Fn + Fg = mv^2/r (Fn cancels out therefore):
Fg = mv^2/r
mg = mv^2/r (the m's cancel out)
g = v^2/r
(g)(r) = v^2
(9.80)(10) = v^2
98 = v^2

there is no friction in the system therefore:
ΔE = 0
(Epf - Epi) + (Ekf - Eki) = 0
Epf - Epi + Ekf - Eki = 0
Epf + Ekf = Epi + Eki (Eki = 0 therefore):
Epf + Ekf = Epi
mghf + 1/2mvf^2 = mghi (the m's cancel out):
ghf + 1/2vf^2 = ghi
(9.80)(2*10) + 1/2(98) = (9.80)hi (for here I put hf = 2R because the cart is at the very top of the loop here so therefore hf = 2 times the radius)
245/9.80 = hi
25m = hi

My question is, could someone please check this question to see if I did it right or if my answer is right, because I have three questions following this one that are quite similar in setup so I just want to make sure I am doing the first one correctly before I move onto the other three.

 

[anathema]

Your approach and calculations seem correct. However, I would suggest labeling your variables and equations more clearly for easier understanding. Also, make sure to include units in your calculations and final answer. Other than that, your answer of 25m for the minimum value of h seems correct. Good job!

 

[thomate1]

I would suggest double checking your calculations and making sure all units are consistent. It seems like you have the right approach, using the centripetal force equation and conservation of energy. However, your final answer of 25m seems too high. I would recommend going through your calculations again and checking for any mistakes. Additionally, it may be helpful to label your variables and units clearly to avoid any confusion. Once you have confirmed your calculations, you can move on to the other questions with confidence.