Answer to Question #179458 in Mechanics | Relativity for Ahmad

Question

A small frictionless car runs down a ramp, and around the inside of a vertical, circular, loop of track. If the car takes a circular path of radius 20 cm as it goes around the loop and the car starts from rest, calculate the minimum starting height, h, on the ramp at which the car must start to complete the loop without losing contact with the track.

Answer 1

Suppose the speed of the car at the topmost point of the loop is v. Taking the gravitational potential energy to be zero at the platform and assuming that the car starts with a negligible speed the conservation of energy shows

$$0=−mgh+\frac{1}{2}mv^2$$

or,

$$mv^2=2mgh$$

where m is the mass of the car. The car moving in a circle must have radial acceleration 

$v^2R$  at this instant. The force on the car are $mg$ due to gravity and N due to the contact with the track. Both these forces are n radial direction at the top o the loop. Thus, from Newton's law,

$$mg+N=\dfrac{mv^2}{R}$$ $$\Rightarrow Mg+N=2mg\frac{h}{R}$$

For h to be minimum N should assume the minimum value which can be zero. Thus,

$$2m\dfrac{g(h_{min})}{R}=mg\\\Rightarrow h_{min}=\dfrac{R}{2}\\\text{Here, R= 20 cm}\\So,\ \ h_{min}=\dfrac{20}{2}=10\ cm$$