A \( 40.0-\mathrm{kg} \) child takes a ride on a Ferris wheel that rotates four times each minute and has a diameter of 18.0 m . a) What is the centripetal acceleration of the child? b) What force (magnitude and direction) does the seat exert on the child at the lowest point of the ride? c) What force does the seat exert on the child at the highest point of the ride? d) What force does the seat exert on the child when the child is halfway between the top and bottom?

To solve the problem, we will break it down into parts and use the known conditions to find the required values. ### Given Data: - Mass of the child, \( m = 40.0 \, \text{kg} \) - Diameter of the Ferris wheel, \( d = 18.0 \, \text{m} \) - Radius of the Ferris wheel, \( r = \frac{d}{2} = 9.0 \, \text{m} \) - Rotations per minute, \( f = 4 \, \text{rev/min} \) ### a) Centripetal Acceleration The formula for centripetal acceleration \( a_c \) is given by: \[ a_c = \frac{v^2}{r} \] where \( v \) is the linear velocity. First, we need to convert the rotations per minute to radians per second: \[ \omega = 2\pi f = 2\pi \left(\frac{4}{60}\right) \, \text{rad/s} \] Then, the linear velocity \( v \) can be calculated as: \[ v = r \cdot \omega \] Now we can calculate \( a_c \). ### b) Force at the Lowest Point At the lowest point, the forces acting on the child are the gravitational force \( F_g \) and the normal force \( F_n \) from the seat. The net force provides the centripetal force: \[ F_n - F_g = m a_c \] where \( F_g = m g \) (with \( g \approx 9.81 \, \text{m/s}^2 \)). We can solve for \( F_n \). ### c) Force at the Highest Point At the highest point, the forces acting on the child are again the gravitational force and the normal force. The equation is: \[ F_n + F_g = m a_c \] We can solve for \( F_n \) here as well. ### d) Force Halfway Between Top and Bottom At this position, the gravitational force acts downward, and the normal force acts upward. The net force is zero since the child is in uniform circular motion: \[ F_n = F_g \] Now, let's perform the calculations step by step. #### Step 1: Calculate \( \omega \) and \( v \) \[ \omega = 2\pi \left(\frac{4}{60}\right) \] \[ v = r \cdot \omega \] #### Step 2: Calculate \( a_c \) \[ a_c = \frac{v^2}{r} \] #### Step 3: Calculate Forces 1. At the lowest point: \[ F_n = m a_c + F_g \] 2. At the highest point: \[ F_n = m a_c - F_g \] 3. Halfway: \[ F_n = F_g \] Now, let's perform the calculations. Calculate the value by following steps: - step0: Calculate: \(2\pi \left(\frac{4}{60}\right)\) - step1: Reduce the fraction: \(2\pi \times \frac{1}{15}\) - step2: Multiply the terms: \(\frac{2}{15}\pi \) - step3: Multiply: \(\frac{2\pi }{15}\) Calculate or simplify the expression \( 9*(2*\pi*(4/60)) \). Calculate the value by following steps: - step0: Calculate: \(9\left(2\pi \left(\frac{4}{60}\right)\right)\) - step1: Remove the parentheses: \(9\times 2\pi \left(\frac{4}{60}\right)\) - step2: Reduce the fraction: \(9\times 2\pi \times \frac{1}{15}\) - step3: Multiply the terms: \(\frac{6}{5}\pi \) - step4: Multiply: \(\frac{6\pi }{5}\) Calculate or simplify the expression \( 40*9.81 \). Calculate the value by following steps: - step0: Calculate: \(40\times 9.81\) - step1: Multiply the numbers: \(392.4\) Calculate or simplify the expression \( (9*(2*\pi*(4/60)))^2/9 \). Calculate the value by following steps: - step0: Calculate: \(\frac{\left(9\left(2\pi \left(\frac{4}{60}\right)\right)\right)^{2}}{9}\) - step1: Remove the parentheses: \(\frac{\left(9\times 2\pi \left(\frac{4}{60}\right)\right)^{2}}{9}\) - step2: Reduce the fraction: \(\frac{\left(9\times 2\pi \times \frac{1}{15}\right)^{2}}{9}\) - step3: Multiply: \(\frac{\left(\frac{6\pi }{5}\right)^{2}}{9}\) - step4: Simplify the expression: \(\frac{\frac{\left(6\pi \right)^{2}}{5^{2}}}{9}\) - step5: Rewrite the expression: \(\frac{\left(6\pi \right)^{2}}{225}\) - step6: Evaluate the power: \(\frac{36\pi ^{2}}{225}\) - step7: Reduce the fraction: \(\frac{4\pi ^{2}}{25}\) Calculate or simplify the expression \( 40*1.579137 + 392.4 \). Calculate the value by following steps: - step0: Calculate: \(40\times 1.579137+392.4\) - step1: Multiply the numbers: \(63.16548+392.4\) - step2: Add the numbers: \(455.56548\) Calculate or simplify the expression \( 392.4 \). Numbers by following steps: - step0: Solution: \(392.4\) Calculate or simplify the expression \( 40*1.579137 - 392.4 \). Calculate the value by following steps: - step0: Calculate: \(40\times 1.579137-392.4\) - step1: Multiply the numbers: \(63.16548-392.4\) - step2: Subtract the numbers: \(-329.23452\) Let's summarize the calculations step by step: ### Step 1: Calculate Angular Velocity \( \omega \) and Linear Velocity \( v \) 1. **Angular Velocity**: \[ \omega = 2\pi \left(\frac{4}{60}\right) \approx 0.4189 \, \text{rad/s} \] 2. **Linear Velocity**: \[ v = r \cdot \omega = 9 \cdot 0.4189 \approx 3.7699 \, \text{m/s} \] ### Step 2: Calculate Centripetal Acceleration \( a_c \) \[ a_c = \frac{v^2}{r} = \frac{(3.7699)^2}{9} \approx 1.5791 \, \text{m/s}^2 \] ### Step 3: Calculate Forces 1. **Gravitational Force**: \[ F_g = m g = 40 \cdot 9.81 \approx 392.4 \, \text{N} \] 2. **Force at the Lowest Point**: \[ F_n = m a_c + F_g = 40 \cdot 1.5791 + 392.4 \approx 455.57 \, \text{N} \] 3. **Force at the Highest Point**: \[ F_n = m a_c - F_g = 40 \cdot 1.5791 - 392.4 \approx -329.23 \, \text{N} \] (This indicates that the normal force is less than the gravitational force, meaning the child would be in free fall at this point.) 4. **Force Halfway Between Top and Bottom**: \[ F_n = F_g \approx 392.4 \, \text{N} \] ### Final Answers: a) The centripetal acceleration of the child is approximately \( 1.58 \, \text{m/s}^2 \). b) The force exerted by the seat on the child at the lowest point is approximately \( 455.57 \, \text{N} \) upward. c) The force exerted by the seat on the child at the highest point is approximately \( -329.23 \, \text{N} \) (indicating free fall). d) The force exerted by the seat on the child when halfway between the top and bottom is approximately \( 392.4 \, \text{N} \) upward.