91Ó°ÊÓ

Torque is a fundamental concept in rotational dynamics, which deals with how forces cause rotational motion. Imagine opening a door; the force applied further from the hinges results in a greater turning effect. This is due to torque. In simple terms, torque is the rotational equivalent of linear force.
The formula for calculating torque is:

• \( \tau = r \times F \)

where \(\tau\) is the torque, \(r\) is the radius or distance from the pivot point to where the force is applied, and \(F\) is the force applied.
In our merry-go-round example, the child applies an 18.0 N force at the edge. With a radius of 2.40 m, the torque is calculated as:

• \[ \tau = 2.40 \, \text{m} \times 18.0 \, \text{N} = 43.2 \, \text{N} \cdot \text{m} \]

This demonstrates how a force applied at a distance can cause rotational movement, known as angular acceleration.

The moment of inertia is a property of any object that rotates, determining how much torque is needed for a desired angular acceleration. It’s akin to mass in linear motion but for rotation. More mass spread out from the axis means a higher moment of inertia, making it harder to spin the object.
The formula to relate torque and angular acceleration is:

• \( \tau = I \alpha \)

where \(I\) is the moment of inertia and \(\alpha\) is the angular acceleration.
Given the moment of inertia of the merry-go-round is \(2100 \, \text{kg} \cdot \text{m}^2\), the angular acceleration can be found as:

• \[ \alpha = \frac{\tau}{I} = \frac{43.2 \, \text{N} \cdot \text{m}}{2100 \, \text{kg} \cdot \text{m}^2} = 0.02057 \, \text{rad/s}^2 \]

A higher moment of inertia means any force applied will result in a smaller angular acceleration compared to a smaller moment of inertia.

In physics, work is done when a force causes displacement. For rotational systems, work involves torque and angular displacement. When the child spins the merry-go-round, they do work to overcome inertia and friction, increasing rotational kinetic energy.
The formula for rotational work is:

• \( W = \tau \theta \)

where \(\theta\) is angular displacement.
Using computed values for torque and \(\theta = 2.314 \, \text{rad}\), the work done is:

• \[ W = 43.2 \, \text{N} \cdot \text{m} \times 2.314 \, \text{rad} = 99.98 \, \text{J} \]

Power, the rate at which work is done, calculated as:

• \( P = \frac{W}{t} = \frac{99.98 \, \text{J}}{15.0 \, \text{s}} = 6.67 \, \text{W} \)

Power informs us about how quickly energy is being used or transferred. Here, it is the average energy per second supplied by the child to spin the merry-go-round.