A dad pushes tangentially on a small hand-driven merry-go-round and is able to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform disk of radius 2.5 m and has a mass of 330 kg, and two children (each with a mass of 25 kg) sit opposite each other on the edge. Calculate the torque required to produce the acceleration, neglecting frictional torque. What force is required at the edge?

Verified step by step guidance

1. Step 1: Calculate the angular acceleration (α) of the merry-go-round. First, convert the final frequency from revolutions per minute (rpm) to radians per second (rad/s). Use the formula: \( \omega = \frac{2\pi f}{60} \), where \( f \) is the frequency in rpm. Then, use the formula for angular acceleration: \( \alpha = \frac{\Delta \omega}{\Delta t} \), where \( \Delta \omega \) is the change in angular velocity and \( \Delta t \) is the time interval.
2. Step 2: Calculate the moment of inertia (I) of the system. The merry-go-round is a uniform disk, so its moment of inertia is given by \( I_{disk} = \frac{1}{2} M R^2 \), where \( M \) is the mass of the disk and \( R \) is its radius. Additionally, the two children sitting on the edge contribute to the moment of inertia as point masses: \( I_{children} = 2 m R^2 \), where \( m \) is the mass of one child. Add these contributions to find the total moment of inertia: \( I_{total} = I_{disk} + I_{children} \).
3. Step 3: Use the relationship between torque (τ), moment of inertia (I), and angular acceleration (α): \( \tau = I \alpha \). Substitute the values of \( I_{total} \) and \( \alpha \) calculated in the previous steps to find the required torque.
4. Step 4: Calculate the force (F) required at the edge of the merry-go-round. Torque is related to force and radius by the formula: \( \tau = F R \). Rearrange this formula to solve for force: \( F = \frac{\tau}{R} \). Substitute the values of \( \tau \) and \( R \) to find the force.
5. Step 5: Summarize the results. The torque required to produce the acceleration is determined in Step 3, and the force required at the edge is determined in Step 4. Ensure all units are consistent and properly labeled for clarity.

Key Concepts

Torque

Torque is a measure of the rotational force applied to an object, calculated as the product of the force and the distance from the pivot point (lever arm). In this scenario, the torque is essential for determining how much rotational force the dad needs to apply to accelerate the merry-go-round. The formula for torque (τ) is τ = r × F, where r is the radius and F is the applied force.

Moment of Inertia

The moment of inertia is a property of a body that quantifies its resistance to rotational acceleration about an axis. For a uniform disk, it is calculated using the formula I = (1/2) m r², where m is the mass and r is the radius. Understanding the moment of inertia is crucial for calculating the angular acceleration of the merry-go-round when torque is applied.

Angular Acceleration

Angular acceleration is the rate of change of angular velocity over time, typically measured in radians per second squared (rad/s²). It can be calculated using the relationship between torque and moment of inertia, expressed as τ = Iα, where α is the angular acceleration. In this problem, determining the angular acceleration is necessary to find the torque required to achieve the desired frequency of rotation.