Question

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

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

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.

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

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).

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)

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.

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).

Key concepts

Centripetal Force

Centripetal force is an essential concept in understanding circular motion. It is the force that keeps an object moving in a circular path, always directed toward the center of the circle.
This force is not a new type of force, but rather the result of other forces, such as gravity or tension, acting on an object in such a way as to cause circular motion.

The magnitude of centripetal force can be calculated using the formula:
\[ F_c = \frac{mv^2}{r} \]

Where:

• \(F_c\) is the centripetal force
• \(m\) is the mass of the object
• \(v\) is the velocity
• \(r\) is the radius of the circular path

When riders on a Ferris wheel travel around the circle, this force ensures they remain on their circular path. Understanding centripetal force helps in recognizing how the forces such as gravity and the normal force balance each other, especially at different points in a circular path, like the top and bottom of a Ferris wheel ride.

Free-Body Diagram

A free-body diagram (FBD) is a useful tool to visualize the forces acting on an object. It simplifies the understanding of how forces interact and balance each other.

When analyzing the motion of a rider on a Ferris wheel, especially at the bottom and top of the circle, drawing a free-body diagram can be very helpful. On the diagram:

• Identify the object of interest, such as the rider.
• Represent all the forces acting on the rider with arrows.
• For the rider on the Ferris wheel, the primary forces to consider are gravitational force (weight) and the normal force from the seat.
• At the bottom of the ride, the gravitational force points downward while the normal force points upward.
• At the top of the ride, both forces act downward. The normal force must decrease to maintain equilibrium in circular motion.

Using a free-body diagram, you can effectively set up equations to solve for unknown forces, like the normal force, by applying Newton's laws.

Normal Force

Normal force is a contact force perpendicular to the surface with which an object interacts. For a Ferris wheel, this force varies as the rider moves up and down along the circular path.

The normal force at any point can be calculated using the balance of forces in the vertical direction, especially using principles of circular motion. At the bottom of the circle, it counteracts the gravitational force and provides the needed centripetal force.

Using the formula:\[ N - mg = \frac{mv^2}{r} \]The normal force \(N\) can be rearranged to:\[ N = mg + \frac{mv^2}{r} \]Here, the normal force \(N\) is larger than the gravitational force \(mg\), and the centrifugal aspect increases it.

At the top of the ride, the scenario changes; the force balances differently:\[ mg - N = \frac{mv^2}{r} \]Thus,\[ N = mg - \frac{mv^2}{r} \]In this situation, the normal force is less than gravitational force.

Understanding these concepts allows one to calculate how tight or gentle the forces need to be to ensure safe and constant motion on the Ferris wheel.