91Ó°ÊÓ

Torque is a measure of how much a force acting on an object causes that object to rotate. It depends on three things: the amount of force, the distance from the pivot point (the axis of rotation), and the angle between the force and the lever arm. In this exercise, the force is the weight of the bucket, which is pulling downward on the pulley.

• The force is calculated as the weight of the bucket: \( F = mg \), where \( m = 1.0 \, \text{kg} \) and \( g = 9.8 \, \text{m/s}^2 \), giving us \( 9.8 \, \text{N} \).
• This force creates a torque \( \tau \) on the pulley because it acts at a distance (the radius of the pulley, \( r = 0.300 \, \text{m} \)) from the axis of rotation.

The formula for torque \( \tau \) is \( \tau = rF \), which balances the torque equation: \( \tau = 0.300 \, \text{m} \times 9.8 \, \text{N} = 2.94 \, \text{Nm} \). Torque causes the pulley to rotate, and understanding this concept helps us analyze rotational dynamics.

The moment of inertia is like the mass of an object, but for rotation. It represents how difficult it is to change the rotational state of the object. The larger the moment of inertia, the harder it is to spin the object around.

• For a uniform disk, like our pulley, the moment of inertia \( I \) is given by \( I = \frac{1}{2} m r^2 \).
• In this case, \( m = 5.00 \, \text{kg} \) and \( r = 0.300 \, \text{m} \), so we calculate \( I = \frac{1}{2} \times 5.00 \, \text{kg} \times (0.300 \, \text{m})^2 = 0.225 \, \text{kg} \cdot \text{m}^2 \).

Think of the moment of inertia as resistance to rotation. A higher moment of inertia means that more torque is needed to start or stop rotating. It's an essential part of solving problems in rotational dynamics, giving us insight into how and why objects resist changes to their rotation.

Angular acceleration is how quickly an object's angular (rotational) velocity changes. It's very similar to linear acceleration, but instead of measuring how fast an object speeds up in a straight line, it measures how quickly it spins faster.

• By using the torque and moment of inertia, we can calculate the angular acceleration \( \alpha \) of the pulley using \( \tau = I \alpha \).
• Rearranging gives us \( \alpha = \frac{\tau}{I} \).

In our example, \( \alpha = \frac{2.94 \, \text{Nm}}{0.225 \, \text{kg} \cdot \text{m}^2} = 13.067 \, \text{rad/s}^2 \). This value indicates how quickly the pulley speeds up its rotation as the bucket descends. Understanding angular acceleration is crucial since it connects the motion to the forces and inertia involved.

Angular displacement refers to how much an object has rotated over a given period. It measures the change in the angle as an object spins.

• Firstly, if the object starts from rest, the initial angular velocity \( \omega_0 \) is 0.
• To find the angular displacement \( \theta \), you can use the formula \( \theta = \omega_0 t + \frac{1}{2} \alpha t^2 \).

In our problem, \( \theta = 0 + \frac{1}{2} \times 13.067 \, \text{rad/s}^2 \times (5.00 \, \text{s})^2 = 163.34 \, \text{rad} \).
This result tells us how far the pulley has turned in radians. Since we often want to know how many times the object has gone around, we convert from radians to rotations. We calculate the number of rotations by dividing by \( 2\pi \), resulting in \( \frac{163.34 \, \text{rad}}{2\pi} \approx 26 \, \text{rotations} \). Knowing angular displacement helps us understand how things move in circles and is pivotal for solving rotational problems.