91Ó°ÊÓ

Torque is fundamental in understanding rotational dynamics. It is essentially the measure of how much a force acting on an object causes that object to rotate. Imagine applying a force on the edge of a playground merry-go-round: if the force is applied further from the center, it causes more rotational motion. The formula to calculate torque (\( \tau \)) is given by:

1. \( \tau = F \cdot r \)

where \( F \) is the force applied, and \( r \) is the distance from the rotation axis to the point where the force is applied. For our merry-go-round, the child applies a force of \(18.0\ \text{N}\) at a radius of \(2.40\ \text{m}\). So, the torque is \(43.2\ \text{N} \cdot \text{m}\). This torque sets the merry-go-round in motion by overcoming its resistance to rotation.

Angular acceleration (\( \alpha \)) refers to how quickly the angular speed of an object is changing. It is analogous to linear acceleration, but in rotational terms. The angular acceleration can be calculated once we know the torque and the moment of inertia through the relation:

1. \( \alpha = \frac{\tau}{I} \)

In this scenario, the torque is \(43.2\ \text{N}\cdot\text{m}\) and the moment of inertia \(I\) is \(2100\ \text{kg} \cdot \text{m}^2\). Plugging in these values, we get an angular acceleration of \(0.02057\ \text{rad/s}^2\). This means that every second, the angular speed increases by this amount, making the merry-go-round spin faster and faster during the child's push.

Moment of inertia (\( I \)) is a crucial concept when dealing with rotational motion. It acts like mass in linear motion, basing how difficult it is to change the angular velocity of an object. Just as it is harder to push a massive box than a light one, it is harder to spin an object with a large moment of inertia. The merry-go-round has a moment of inertia of \(2100\ \text{kg} \cdot \text{m}^2\).

• A higher moment of inertia means the object is harder to spin.
• A lower moment of inertia makes it easier to spin the object.

The value of \(2100\ \text{kg} \cdot \text{m}^2\) indicates the resistance of the merry-go-round to angular acceleration and demonstrates the effect of mass distribution away from the rotation axis.

The Work-Energy Principle in rotation states that work done by external forces on a rotating system results in a change in the rotational energy of the system. In simpler terms, when the child pushes the merry-go-round, they are doing work, which then transforms into rotational energy, causing the merry-go-round to spin. The work done (\( W \)) by the child can be calculated using:

1. \( W = \frac{1}{2} I \omega^2 \)

where \( \omega \) is the angular speed after the force application. For our merry-go-round, given \( \omega = 0.30855\ \text{rad/s} \), the work done by the child is approximately \(99.45\ \text{J}\). This energy transition is key in imparting motion, translating the child's physical effort into the rotation of the merry-go-round.

Average power (\( P \)) is defined as the rate at which work is done or energy is transferred over time. It provides insight into how quickly energy is being consumed or transferred. Mathematically, it is given by:

1. \( P = \frac{W}{t} \)

where \( W \) is the work done and \( t \) is the time interval. In our situation, with \( W = 99.45\ \text{J} \) and \( t = 15.0\ \text{s} \), the average power supplied by the child is \(6.63\ \text{W}\). This tells us how efficiently or intensely the child adds rotational energy to the merry-go-round within a given time span.