91Ó°ÊÓ

Torque can be thought of as the rotational equivalent of linear force. When we apply a torque to an object, it tends to produce rotational motion. Just like force can cause an object to accelerate, torque causes an object to experience angular acceleration. In our scenario, we are applying equal torques to both a hollow cylinder and a solid sphere. But what is this mysterious torque?Torque is defined mathematically as the product of force and the distance from the pivot point at which the force is applied. If you think about using a wrench, the torque is greater if you apply the same force farther from the nut. This is expressed as:\[ \tau = r \, F \]where:

• \(\tau\) is the torque,
• \(r\) is the distance from the axis of rotation (the lever arm), and
• \(F\) is the force applied.

In the exercise, both objects experience the same torque, which means the same rotational force is trying to spin the cylinder and the sphere. But their responses, described by their angular accelerations and angular speeds, depend on their moments of inertia, which we'll explore next.

Angular acceleration is a measure of how quickly an object changes its angular speed. It is similar to linear acceleration, but applies to rotational movement. When you apply a torque to an object, its angular acceleration depends on the torque and the object's moment of inertia (which is its resistance to rotation).Angular acceleration is represented by the Greek letter \( \alpha \) and is determined by the equation:\[ \alpha = \frac{\tau}{I} \]where:

• \( \alpha \) is the angular acceleration,
• \( \tau \) is the torque applied, and
• \( I \) is the moment of inertia.

From the exercise, since the solid sphere has a smaller moment of inertia compared to the hollow cylinder, it means with the same torque, the sphere will have a larger angular acceleration. This happens because the sphere resists changes in its rotational state less than the cylinder. This principle, that objects with lower moments of inertia experience greater angular accelerations under the same torque, is key to understanding rotational dynamics.

Angular speed is how fast an object spins around its axis and is usually denoted by the Greek letter \( \omega \). It's akin to how fast an object might move in a straight line, except here it's about rotation. In our exercise scenario, after applying equally strong torques to the cylinder and the sphere, we want to see which one ends up spinning faster over a period of time.The relationship between angular speed and angular acceleration can be expressed through time as:\[ \omega = \alpha t \]where:

• \( \omega \) is the final angular speed,
• \( \alpha \) is the angular acceleration, and
• \( t \) is the time for which torque is applied.

With the sphere having greater angular acceleration (as derived from its smaller moment of inertia), it will reach a higher angular speed than the cylinder when the same torque is applied for the same duration. This relationship becomes evident because the smaller the resistance to rotation, the faster an object spins up, showing that less force is needed to reach a higher speed. This concept is crucial in systems where speed and efficiency are desired.