A playground merry-go-round has radius 2.40 m and moment of inertia 2100 kg•m^2 about a vertical axle through its center, and it turns with negligible friction. A child applies an 18.0-N force tangentially to the edge of the merry-go-round for 15.0 s. If the merry-go-round is initially at rest (b) How much work did the child do on the merry-go-round?

Verified step by step guidance

1. Calculate the torque applied by the child. Torque (\(\tau\)) can be calculated using the formula \(\tau = r \times F\), where \(r\) is the radius of the merry-go-round and \(F\) is the force applied.
2. Determine the angular acceleration (\(\alpha\)) using the formula \(\alpha = \frac{\tau}{I}\), where \(I\) is the moment of inertia of the merry-go-round.
3. Calculate the angular displacement (\(\theta\)) using the kinematic equation for rotational motion \(\theta = \frac{1}{2} \alpha t^2\), where \(t\) is the time during which the force is applied.
4. Compute the work done by the child using the work-energy theorem for rotational motion, which states that the work done (\(W\)) is equal to the change in rotational kinetic energy. Since the merry-go-round starts from rest, the initial kinetic energy is 0, and the work done can be calculated as \(W = \frac{1}{2} I \omega^2\), where \(\omega\) is the final angular velocity.
5. To find \(\omega\), use the relationship \(\omega = \alpha t\), and substitute this into the equation for work done to express \(W\) in terms of \(\alpha\), \(I\), and \(t\).

Key Concepts

Work

Work is defined as the product of the force applied to an object and the distance over which that force is applied, in the direction of the force. Mathematically, it is expressed as W = F × d × cos(θ), where θ is the angle between the force and the direction of motion. In this scenario, since the force is applied tangentially, θ is 0 degrees, simplifying the equation to W = F × d.

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). It is given by the formula τ = r × F, where τ is torque, r is the radius, and F is the force applied. In the case of the merry-go-round, the child’s force creates torque that causes it to rotate.

Moment of Inertia

Moment of inertia is a property of a body that quantifies its resistance to angular acceleration about a given axis. It depends on the mass distribution relative to the axis of rotation and is calculated using I = Σ(m × r²), where m is mass and r is the distance from the axis. For the merry-go-round, the moment of inertia affects how much torque is needed to achieve a certain angular acceleration when the child applies force.