A roller coaster car of mass 1000kg starts from rest and rolls down a track from

Question

A roller coaster car of mass 1000kg starts from rest and rolls down a track from a height of 20-m. the diameter of the loop is 10m. ignore friction and air resistance.
Find
A The lowest speed that the car needs to reach the top of the loop.
B The force exerted on the car by the track at the top of the loop.
C From what minimum height above the bottom of the loop can the car be released so that it does not lose contact with the track at the top of the loop? (Hint: for minimum speed the normal force is zero)

Explanation / Answer

According to conservation of energy,
1/2mv^2 = mgh
v = 2gh = 19.79 m/s

the speed of the roller at bottem is 19.79 m/s
net force to the center = N + mg

(A) If we wanted to calculate the minimum or critical velocity needed for the block to just be able to pass through the top of the circle without the rope sagging then we would start by letting the tension in the rope approaches zero.

0 = m(v2/r) - mg
m(v2/r) = mg
v2/r = g
v2 = rg
v = (rg)

V = 10*9.8 = 9.89 m/s

The lowest speed that the car needs to reach the top of the loop is 9.89 m/s

(B)

the force axerted on car is N = mg + 1/2mv^2

                                           N = 980+49000

                                              = 49980 N

The force exerted on the car by the track at the top of the loop is 49980 N.

(C) the minimum hight,

     V^2 = 2gh = rgf

            h = r/2 = 5 m

the minimum height above the bottom of the loop can the car be released so that it does not lose contact with the track at the top of the loop is 5m.

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