91Ó°ÊÓ

Torque is a measure of the rotational effect of a force applied to a rotating object, like the pulley in our exercise. Imagine you're pushing open a door. The farther from the hinges you push, the easier it is to turn the door. This is what torque does—it depends on two things: the force applied and how far that force is from the axis of rotation. This distance is known as the lever arm or radius.

In mathematical terms, torque \( \tau \) is calculated with the formula \( \tau = F \cdot r \), where \( F \) is the force applied, and \( r \) is the radius. It is measured in newton-meters (Nm).

• For our pulley, a force of 4.2 N applied at a 0.1-meter radius resulted in a torque of 0.42 Nm.

Torque is crucial in determining how quickly an object will start rotating when a force is applied. It's the rotational equivalent of force in linear motion.

Angular acceleration \( \alpha \) describes how quickly the angular speed of an object changes over time. It's like the car's accelerator pedal for rotating objects, telling us how speed increases as time goes by. If you spin a wheel faster and faster, you're increasing its angular acceleration.

Angular acceleration is governed by Newton's second law for rotation, which states that torque \( \tau \) equals rotational inertia \( I \) times angular acceleration \( \alpha \): \( \tau = I \alpha \). Therefore, \( \alpha = \frac{\tau}{I} \).

• Using the exercise values, with \( \tau = 0.42 \text{ Nm} \) and \( I = 1.0 \times 10^{-3} \text{ kg} \cdot \text{m}^2 \), we calculated \( \alpha = 420 \text{ rad/s}^2 \).

This tells us how fast the pulley gains speed. The higher the angular acceleration, the quicker it spins.

Angular speed \( \omega \) measures how fast an object rotates or spins. It’s similar to linear speed but for circular paths. If you think about the second hand of a clock, it's the number of tick marks it passes per second. It's measured in radians per second (rad/s).

To determine the angular speed of an object, you can use the rotational kinematic equation: \( \omega = \omega_0 + \alpha t \), where \( \omega_0 \) is the initial angular speed, \( \alpha \) is the angular acceleration, and \( t \) is time.

• In our problem, as the pulley started from rest, \( \omega_0 \) was 0. Using \( \alpha = 420 \text{ rad/s}^2 \) and \( t = 3 \text{ s} \), the angular speed was calculated to be \( 1260 \text{ rad/s} \).

Understanding angular speed helps us track how quickly the object completes its circular paths over time, an essential factor in many engineering and mechanical applications.