Flywheel Rotational Dynamics

Part a

6 minutes 5 marks

Question

A flywheel, initially at rest, accelerates uniformly to an angular velocity of 30 rad s⁻¹ in 5.0 s. The flywheel is a solid disk with mass 10 kg and radius 0.20 m. The moment of inertia of a solid disk is \(I = ½ MR²\).

(i) Calculate the angular acceleration of the flywheel. [1]

(ii) Calculate the moment of inertia of the flywheel. [1]

(iii) Determine the resultant torque required to produce this acceleration. [1]

(iv) Calculate the rotational kinetic energy of the flywheel when it reaches the angular velocity of 30 rad s⁻¹. [2]

Hint Hint

Hint

Use rotational kinematic equations for (i), the given formula for (ii), Newton’s second law for rotation (τ = Iα) for (iii), and the formula for rotational kinetic energy for (iv).

Answer Answer

Answer Answer

Complete Answer

(i) 6.0 rad s⁻²
(ii) 0.20 kg m²
(iii) 1.2 N m
(iv) 90 J

Part b

5 minutes 4 marks

Question

The flywheel is now disconnected from the motor and a constant frictional torque is applied, bringing it to rest in 20 s.

(i) Calculate the magnitude of the angular deceleration. [1]

(ii) Determine the magnitude of the frictional torque. [1]

(iii) Calculate the total angular displacement of the flywheel during the deceleration period. [2]

Hint Hint

Hint

Use rotational kinematics for (i) and (iii). Use \(\tau = I\alpha\) for (ii). Remember the final angular velocity is zero.

Answer Answer

Answer Answer

Complete Answer

(i) 1.5 rad s⁻²
(ii) 0.30 N m
(iii) 300 rad

Part c

3 minutes 2 marks

Question

Outline how conservation of angular momentum applies when the radius of a rotating object changes, assuming no external torque acts. [2]

Hint Hint

Hint

Start with the definition of angular momentum (L = Iω). State the condition for conservation (no external torque). Explain how I changes with radius and what must happen to ω.

Answer Answer
Mark Scheme Mark Scheme
Explanation Explanation

Answer Answer

Complete Answer

Angular momentum \(L = I\omega\) is conserved if no net external torque acts. If the moment of inertia \(I\) changes (e.g., due to radius change, since \(I\) often depends on \(R^2\)), the angular velocity \(\omega\) must change inversely (e.g., if \(I\) decreases, \(\omega\) increases) to keep \(L\) constant.

Mark Scheme Mark Scheme

Answer Explanation Answer Explanation