A roller coaster car may be approximated by a block of mass m. As per your desig

Question

A roller coaster car may be approximated by a block of mass m. As per your design, you want the car to have a speed of 3 m/s at the height h above the ground just before it begins to slide down along the frictionless track. The car is to encounter a loop of radius R, as shown (note that the car will be traveling on the inner part of the circular loop NOT on the outside/above the circular loop). Assume that the initial height h is great enough so that the car never loses contact with the track. Find an expression for the kinetic energy of the car at the top of the loop. Find the minimum initial height h at which the car can be released that still allows the car to stay in contact with the track at the top of the loop (i.e. just barely remain in contact with the track at the top).

Explanation / Answer

b)

here,

centripetal force Fc=m*g

m*v^2/R=m*g

===> v^2=g*R

now,

by using energy relatuon,

m*g*h_min=m*g*(2*R)+1/2*m*v^2

g*h_min=g*(2*R)+1/2*v^2

g*h_min=g*(2*R)+1/2*(g*R)

h_min=(2*R)+1/2*R)

h_min=(5/2)*R ----is answer

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