Calculating the Minimum Height for a Safe Amusement Park Ride: A Physics Problem

[courtrigrad]

A car in an amusement park ride rolls without friction around the track. It starts from rest at point A at a height h above the bottom of the loop. (a) What is the minimum value of h (in terms of R) such that the car moves around the loop without falling off at the top (point B)? (b) If $$h = 3.5R$$ and $$R = 30 m$$ compute the speed, radial acceleration, and tangential acceleration of the passengers when the car is at point C, which is at the end of the horizontal diameter. Show acceleration components in a diagram.

All I know is that at point A, the kinetic energy will be equal to 0. Do I have to use $$K_{1} + U_{1} + W_{other} = K_{2} + U_{2}$$? Also does $$h = 2R$$

Any help would be appreciated

Thanks

 

[kp]

yes, and at point B you will have kenetic and potential energy.

does h = 2r? no

 

[andrevdh]

Yes the mechanical energy at A will be equal to the mechanical energy at B of the car. At B the car will need some force to supply the centripetal acceleration. Its weight is a candidate - if it moves faster the track will need to supply an additional normal force to increase the needed centripetal force. When this normal force is zero at the top the car is just leaving the track. Therefore the minimum centripetal acceleration at the top will be g. Use the connection between speed and centipetal acceleration to determine what its minimum speed will be a the top.

 

[PhY_InTelLecT]

In this case, the car must have enough PE to convert into KE at the bottom of the loop and back into PE as it moves around the loop.. Therefore:
$$change PE_i= change KE_f= change PE_f$$
simplifying it, $$change PE_i= change PE_f$$, provided there is no energy loses, if there is, then add it in.. since you want the h to be as small as possible, we can assume that the car actually loses all its KE at the top of the loop. So all you have to do is to sub in the values and get the h at start.