A roller-coaster car may be represented by a block of mass 50.0kg . The car is r

Question

A roller-coaster car may be represented by a block of mass 50.0kg . The car is released from rest at a height h = 51.0m above the ground and slides along a frictionless track. The car encounters a loop of radius R = 17.0m at ground level, as shown. As you will learn in the course of this problem, the initial height 51.0m is great enough so that the car never loses contact with the track.

Find an expression for the kinetic energy K of the car at the top of the loop.

Find the minimum initial height hmin 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.

Explanation / Answer

K=m*g*(h-2*R)

for h_min, the centripetal acceleration must equal gravity at the top of the loop.
v^2/R=g
or
v^2=g*R

Using the energy equation from above

.5*m*v^2=m*g*(h_min-2*R)

combine
.5*m*g*R=m*g*(h_min-2*R)

solve for h_min
h_min=2.5*R

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