Question

A rollercoaster is traveling at a speed of 12.0 m/s when it enters a downwardportion of the track shaped like a circle of radius 15.0 m. Neglect friction betweenthe rollercoaster and the track.(a)Draw a free-body diagram for a rider inside the roller coaster when she is atan arbitrary angle ? on the circular portion of the track. Approximate the rider as ablock sitting on a surface with sufficient friction to keep her from sliding.(b)At what angle?does the normal force on the rider equal half of her weight?

Answer 1

PART A

The normal force is applied by the seat on the person towards the center, the centrifugal force is away from the center and the normal component of the weight is .

PART B

We are given that the normal force at particular angle is halg of the weight, so we get,

The force balance gives us:

.........EQ1

where is the final velocity at the angle and is the radius of the circular path.

We know that the energy will be conserved, so the initial total energy will be only kinetic energy (we are considering the initial potential energy is zero) with initial velocity . But when the system reaches the angle the kinetic energy will decrease, but potential energy will be gained. Since, there is nor friction between the surface and the roller coaster, so

.....EQ2

Now the initial kinetic energy will be:

......EQ3

If the center of mass rises up by height 'h' then the height will be

Thus the kinetic and the potential energy at angle will be:

......EQ4

Substituting EQ3 and EQ4 in EQ2, we get

..................EQ5

Equating EQ5 and EQ1 we get

.............EQ6

substituting in EQ6 we get