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

Question

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

Find the kinetic energy K of the car at the top of the loop. Express your answer numerically, in joules.

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. Express your answer numerically, in meters.

Explanation / Answer

Given that

mass m=50 kg

radius r=17 m

initial height hi=51 m

now we find the kinetic energy

the kinetic energy KE=m*g*(h-2*R)=50*9.8(51-2*17)=8330 J

now we find the minimum height

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 =2.5*17=42.5 J

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