A 34.1 kg child rides a Ferris wheel whose diameter is 18.3 m. When the ride is turning at a constant rate, the child’s speed has a constant value of 4.23 m/s. Find the normal force the seat exerts on the child at:

a) The top of the ride
b) The bottom of the ride

a= v²/R.


top

ma= mg –N
N=m(g-a) =
=m (g- v²/R) =
= 34.1{9.8 - (4.23²/9.15 )}=
=267 N.

bottom

ma=N-mg
N= m(g+a) =
= m (g+ v²/R) =
= 34.1{9.8 +(4.23²/9.15 )}=
=401 N

To find the normal force exerted on the child at the top and bottom of the ride, we can use the principles of circular motion.

The normal force is the force exerted by a surface perpendicular to it. In this case, the seat exerts the normal force on the child.

Let's start with the top of the ride.

a) At the top of the ride, the child is moving in a circular path with a constant speed. This means there must be a net force acting towards the center of the circle to keep the child moving in a circle.

The forces acting on the child at the top are:
1. Gravitational force acting downward (mg), where m is the mass of the child and g is the acceleration due to gravity.
2. Normal force exerted by the seat, pointing towards the center of the circle.

Since the child is not moving vertically, the net force in the vertical direction is zero. This means the normal force is equal in magnitude to the gravitational force.

First, let's find the gravitational force (mg):
mass (m) = 34.1 kg
acceleration due to gravity (g) = 9.8 m/s^2

Gravitational force (mg) = (34.1 kg) * (9.8 m/s^2) = 334.18 N

Therefore, at the top of the ride, the normal force is also 334.18 N. The seat exerts a force of 334.18 N towards the center of the circle to keep the child moving in a circular path.

Now, let's move on to the bottom of the ride.

b) At the bottom of the ride, the child is still moving in a circular path with a constant speed, but the direction of the net force changes. The net force must now be directed towards the center of the circle, which means the normal force must act upward.

The forces acting on the child at the bottom are:
1. Gravitational force acting downward (mg),
2. Normal force exerted by the seat, pointing towards the center of the circle, but opposite to the gravitational force.

Again, the net force in the vertical direction is zero. This means the normal force at the bottom of the ride equals the gravitational force (mg).

Using the same values for mass (m) and acceleration due to gravity (g) as before:

Gravitational force (mg) = (34.1 kg) * (9.8 m/s^2) = 334.18 N

Therefore, at the bottom of the ride, the normal force is also 334.18 N. The seat exerts a force of 334.18 N towards the center of the circle to keep the child moving in a circular path.