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

Verified step by step guidance

1. First, understand that work done on an object is calculated using the formula: \( W = F \cdot d \cdot \cos(\theta) \), where \( F \) is the force applied, \( d \) is the distance over which the force is applied, and \( \theta \) is the angle between the force and the direction of motion. In this case, the force is tangential, so \( \theta = 0 \) degrees, and \( \cos(0) = 1 \).
2. Next, calculate the distance \( d \) that the edge of the merry-go-round travels. Since the force is applied tangentially, the distance can be found using the arc length formula: \( d = r \cdot \Delta\theta \), where \( r \) is the radius of the merry-go-round and \( \Delta\theta \) is the angular displacement in radians.
3. To find \( \Delta\theta \), use the relationship between angular acceleration \( \alpha \), time \( t \), and angular displacement: \( \Delta\theta = \frac{1}{2} \cdot \alpha \cdot t^2 \). First, calculate \( \alpha \) using the formula \( \alpha = \frac{\tau}{I} \), where \( \tau \) is the torque and \( I \) is the moment of inertia. Torque \( \tau \) is given by \( \tau = F \cdot r \).
4. Substitute the values into the torque formula: \( \tau = 18.0 \text{ N} \cdot 2.40 \text{ m} \). Then, use \( \alpha = \frac{\tau}{2100 \text{ kg} \cdot \text{m}^2} \) to find the angular acceleration.
5. Finally, substitute \( \alpha \) and \( t = 15.0 \text{ s} \) into \( \Delta\theta = \frac{1}{2} \cdot \alpha \cdot t^2 \) to find the angular displacement. Use this \( \Delta\theta \) to calculate \( d = r \cdot \Delta\theta \), and then substitute \( F \), \( d \), and \( \cos(0) = 1 \) into the work formula \( W = F \cdot d \cdot \cos(\theta) \) to find the work done by the child.

Key Concepts

Moment of Inertia

Moment of inertia is a measure of an object's resistance to changes in its rotation. It depends on the mass distribution relative to the axis of rotation. For a merry-go-round, it quantifies how difficult it is to change its rotational speed, given its mass and shape.

Torque

Torque is the rotational equivalent of force, causing an object to rotate around an axis. It is calculated as the product of force and the radius at which the force is applied. In this scenario, the child's tangential force creates torque, initiating the merry-go-round's rotation.

Work-Energy Principle

The work-energy principle states that work done on an object results in a change in its energy. For rotational motion, work is the product of torque and angular displacement. The child's work on the merry-go-round increases its rotational kinetic energy, starting from rest.