91Ó°ÊÓ

Angular acceleration (\( \alpha \)) is a measure of how quickly an object speeds up or slows down its rotational motion. It is analogous to linear acceleration in straight-line motion.
The unit of angular acceleration is radians per second squared (\( \text{rad/s}^2 \)). In the context of the pulley system, angular acceleration describes how the pulley's rotation changes as the block falls.
The relationship between linear acceleration (\( a \)) and angular acceleration is given by the formula:

• \( a = \alpha r \)

Here, \( r \) is the radius of the pulley. This equation helps in linking the linear fall of the block with the spinning motion of the pulley. If the linear acceleration is known, this formula can easily determine the angular acceleration.
Understanding this relationship is crucial because it bridges the physical connection between different types of motion involved.

Newton's Second Law is a fundamental principle that explains how the velocity of an object changes when it is subject to an external force. The law can be expressed as:

• \( F = ma \)

In this formula, \( F \) is the total force applied to the object, \( m \) is the object's mass, and \( a \) is the linear acceleration.
When applied to the block in the pulley system, Newton's Second Law helps determine the forces at play. The main forces on the block are gravitational force (\( mg \)) pulling it down and the tension (\( T \)) in the cord pulling upwards.
Consequently, Newton's Second Law helps form the equation:

• \( mg - T = ma \)

This shows the relationship between the acceleration of the block and the tension in the cord. Solving this equation is essential for finding the value of the tensile force in the cord which is required for the pulley to rotate.

Torque (\( \tau \)) is a measure of the rotational force applied to an object causing it to rotate around an axis. It can be thought of as the equivalent of force in a linear system but for rotational movement.
The mathematical expression for torque is:

• \( \tau = Tr \)

where \( T \) is the tension force in the cord and \( r \) is the radius of the pulley.
This torque results in angular acceleration based on the moment of inertia. For the pulley system, this is expressed by the equation:

• \( \tau = I\alpha \)

Torque is crucial for calculating how much tension is required to cause a specific angular acceleration in the pulley. Without this torque, the pulley would not be able to rotate, and therefore the block would not fall.

Moment of inertia (\( I \)) refers to an object's resistance to change in its rotational motion. It is the rotational equivalent of mass in linear motion and depends on both the object's mass and the distribution of that mass around the axis of rotation.
For the pulley in this exercise, the given moment of inertia is \( 1.1 \times 10^{-3} \text{ kg} \cdot \text{m}^2 \). This value indicates how much effort in terms of torque is needed to change the pulley's rotational velocity.
In the formula:

• \( I\alpha \)

moment of inertia works alongside torque to determine the angular acceleration. The greater the moment of inertia, the more torque required for the same angular acceleration.
This concept is crucial as it affects the calculations for both the angular acceleration of the pulley and the resulting tension in the cord.