Question

While at the county fair, you decide to ride the Ferris wheel.Having eaten too many candy apples and elephant ears, you find themotion somewhat unpleasant. To take your mind off your stomach, youwonder about the motion of the ride. You estimate the radius of thebig wheel to be 15 ${\rm m}$ , and you use your watch tofind that each loop around takes 25 ${\rm s}$ .
a.What is your speed?
b.What is the magnitude of your acceleration?
c.What is the ratio of your weight at the top of the ride to yourweight while standing on the ground?
d.What is the ratio of your weight at the bottom of the ride toyour weight while standing on the ground?

Answer 1

Concepts and reason

The main concepts used to solve the problem are Newton’s second law, weight, relation between velocity, radius, and angular velocity, and centripetal acceleration.

Initially, use the relation between velocity, radius, and angular velocity to calculate the speed of wheel and use expression of centripetal acceleration to calculate the magnitude of acceleration. Later, use the Newton’s second law at the top of the circle and divide the equation by the true weight to calculate the ratio of your weight at the top of the ride to your weight while standing on the ground. Finally, use the Newton’s second law at the bottom of the circle and divide the equation by the true weight to calculate the ratio of your weight at the bottom of the ride to your weight while standing on the ground.

Fundamentals

The speed of the wheel is expressed as follows:

Here, is the radius of the circle, is the angular velocity, and is the velocity of wheel.

The magnitude of acceleration is expressed as follows:

Here, is the acceleration, is the velocity, and is the radius of the circle.

The Newton’s second law states that the net force is directly proportional to the acceleration with constant of proportionality is mass.

Here, is the mass, a is the acceleration, and is the force.

The weight force is expressed as follows:

Here, m is the mass, g is the acceleration due to gravity, and is the weight.

(A)

Calculate the speed of the wheel.

The speed of the wheel is expressed as follows:

Here, is the radius of the circle, is the angular velocity, and is the velocity of wheel.

Substitute for in expression .

Substitute for , for in expression .

Round off to two significant figures.

The velocity is 3.77 m/s.

(B)

Calculate the magnitude of acceleration.

The magnitude of acceleration is expressed as follows:

Here, is the acceleration, is the velocity, and is the radius of the circle.

Substitute for and for r in expression .

Round off to three significant figures.

The acceleration is .

(C)

Calculate the ratio of your weight at the top of the ride to your weight while standing on the ground.

Apply the Newton’s second law at the top of the circle.

Here, is true weight, N is the normal force, m is the mass, and a is the acceleration.

Substitute for a in expression .

Divide the equation by .

Substitute mg for in expression .

Substitute 3.768 m/s for , for r , and for g in expression .

(D)

Calculate the ratio of your weight at the bottom of the ride to your weight while standing on the ground.

Apply the Newton’s second law at the top of the circle.

Here, is true weight, N is the normal force, m is the mass, and a is the acceleration.

Substitute for a in expression .

Divide the equation by .

Substitute mg for in expression .

Substitute 3.768 m/s for , for r , and for g in expression .

Round off to two significant figures. The ratio is 1.1.

Ans: Part A

The speed is .

Part B

The magnitude of your acceleration is .

Part C

The ratio of your weight at the top of the ride to your weight while standing on the ground is .

Part D

The ratio of your weight at the bottom of the ride to your weight while standing on the ground is .