Final answer:
The magnitude of angular acceleration of a pulley can be derived using the relationship τ = Ia, where torque τ equals force F times radius r from the pivot, and I is the moment of inertia. Angular acceleration is then τ/I and is positive for counterclockwise acceleration.
Step-by-step explanation:
To derive an expression for the magnitude of the angular acceleration a of a pulley, we use the relationship between torque (τ), moment of inertia (I), and angular acceleration (a). By Newton's second law for rotation, the torque is equal to the moment of inertia times the angular acceleration (τ = Ia).
If a force F is applied at a radius r from the pivot, the torque is τ = Fr. Knowing that linear acceleration a is related to angular acceleration by the expression a = rα, we can substitute to find F = ma = m(rα), so τ = m(rα)r = mr²α.
Therefore, the angular acceleration can be expressed in terms of the torque and moment of inertia as a = τ/I.
It's important to remember the convention that counterclockwise motion is considered positive which would imply a positive angular acceleration when an object initially at rest or moving clockwise begins to rotate counterclockwise, or when an object rotating counterclockwise speeds up.
The magnitude of angular acceleration is influenced by the distance from the pivot (r) the force is applied and the mass of the object (m), as well as its moment of inertia. Specifically, angular acceleration is inversely proportional to mass and directly proportional to the force when all else is constant.