Question

1 A roller coaster starts from rest at A, rolls down the track to B, describes a circular loop of 12-m diameter, and travels up and down past point E. Determine the range of values of h for which the roller coaster will not leave the track at D or E. Assume no energy loss due to friction.

Answer 1

Answer:

15m

Explanation:

Given that a roller coaster starts from rest at A, rolls down the track to B, describes a circular loop of 12-m diameter, and travels up and down past point E. Determine the range of values of h for which the roller coaster will not leave the track at D or E. Assume no energy loss due to friction

Solution

At point A

The maximum potential energy = maximum K.E

At point A, the total energy = maximum P.E.

Down the track to point B, the P.E will be converted to maximum K.E.

Hence,

Mgh = 1/2mv^2

Also, the total energy at the roller coaster will be P.E + K.E

I.e mg2r + 1/2mv^2

Where 2r = height of the loop = diameter of the loop.

Since the energy is always conserved, hence

Mgh = mg2r + 1/2mv^2

Let also consider the centripetal acceleration to keep the object in the circle.

F = mV^2 / r = mg

Mass will cancel out

U^2 = rg

Substitute that in the last equation

Mgh = mg2r + 1/2mgr

mgh = mg ( 2r + 1/2r )

Mg will cancel out

h = 2.5r

Where r = 12/2 = 6

h = 2.5 × 6

h = 15m

Therefore, the values of h for which the roller coaster will not leave the track at D or E is 15m.