Toy car executing a vertical loop

[Jared C]

Homework Statement

A small toy car slides down an elaborate track. At one point during this trip, the car will go around a vertical loop with radius 15cm as shown. What is the minimum speed the car must have at the bottom of the loop in order to make it all the way around the loop without falling off?

Homework Equations

K_i + U_i + W_NC = K_f + U_f
K = (1/2)mv²
U_i = mgh
W = Fdcosx
F = ma
a_c = v²/r

The Attempt at a Solution

I tried using the work-energy theorem, using (1/2)mv² = mgh to find the velocity needed for the car to have zero potential energy at the top of the loop, but I know this problem also involves centripetal acceleration and that the car would also need to have some velocity at the top of the loop to finish going around it (zero velocity would cause it to fall downward), but I have no clue how to use this to solve the problem

 

[TomHart]

The key to this problem is that when the car reaches the top of the loop, it is at the speed where it is just at the point where it starts to lose contact with the ramp. In other words, the normal force from the track is equal to 0. So at the bottom of the ramp, you could define that as the point where all of the car's energy is kinetic energy (0 potential energy). So you would have to calculate what speed it would be where the car just starts to lose contact at the top of the loop and work from there. I hope that helps.

Welcome to Physics Forums.

 

[haruspex]

zero velocity would cause it to fall downward

Quite so.
Draw a free body diagram for it when at the top of the loop. What forces act on it? What is the net force? What acceleration needs to result?