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Chapter 9: Problem 56

A particular Ferris wheel takes riders in a vertical circle of radius \(9.0 \mathrm{~m}\) once every \(12.0 \mathrm{~s} .\) a) Calculate the speed of the riders, assuming it to be constant. b) Draw a free-body diagram for a rider at a time when she is at the bottom of the circle. Calculate the normal force exerted by the seat on the rider at that point in the ride. c) Perform the same analysis as in part (b) for a point at the top of the ride.

Short Answer

Expert verified
Answer: The constant speed of the riders is (3/2) * pi m/s. The normal force exerted by the seat on the rider at the bottom point is 9.81m + 4.5m(pi^2), and the normal force exerted by the seat on the rider at the top point is 9.81m - 4.5m(pi^2).

Step by step solution

01

Calculate the speed of the riders

To calculate the speed of the riders, we will use the formula for the circumference of a circle and the given time it takes for one rotation. The formula for the circumference (C) of a circle is: C = 2 * pi * r where r is the radius of the circle (9.0 m in this case). Since it takes the Ferris wheel 12.0 s to complete one rotation, we can find the constant speed (v) of the riders by dividing the circumference by the time: v = C / T Now let's plug in the values to find the speed.
02

Find the speed

Given the radius (r) of the Ferris wheel is 9.0 m, and the time for one rotation (T) is 12.0 s, we can calculate the circumference (C) and constant speed (v) as follows: C = 2 * pi * 9.0 = 18 * pi m v = (18 * pi) / 12.0 = (3/2) * pi m/s
03

Analyze the forces at the bottom point

At the bottom point of the Ferris wheel, the rider experiences two forces: gravitational force (weight) acting downward and normal force exerted by the seat acting upward. The net force toward the center of the circle is equal to the centripetal force needed to keep the rider moving in a circle. Thus, we can write the force balance equation for the vertical (y) direction: N - mg = mv^2 / r We are given the mass (m) of the rider and the value of g (gravitational acceleration) is 9.81 m/s^2. We already calculated the speed (v) in step 2. We can now solve for the normal force (N).
04

Calculate the normal force at the bottom point

Using the force balance equation in Step 3, we can calculate the normal force (N) at the bottom point: N = mg + mv^2 / r Plugging in the given values for m, g, v, and r: N = m(9.81 + ((3/2) * pi)^2 / 9.0) N = 9.81m + 4.5m(pi^2)
05

Analyze the forces at the top point

At the top point of the Ferris wheel, the rider still experiences gravitational force (weight) acting downward. However, the normal force exerted by the seat now also acts downward. The net force toward the center of the circle remains equal to the centripetal force. Thus, the force balance equation for the vertical (y) direction at the top point is: mg - N = mv^2 / r We can now solve for the normal force (N) at the top point.
06

Calculate the normal force at the top point

Using the force balance equation in Step 5, we can calculate the normal force (N) at the top point: N = mg - mv^2 / r Plugging in the given values for m, g, v, and r: N = m(9.81 - ((3/2) * pi)^2 / 9.0) N = 9.81m - 4.5m(pi^2) To sum up, we have calculated the constant speed of the riders as (3/2) * pi m/s, the normal force exerted by the seat on the rider at the bottom point as 9.81m + 4.5m(pi^2), and the normal force exerted by the seat on the rider at the top point as 9.81m - 4.5m(pi^2).

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