Dynamics of Rotational Motion

Practice Questions:

Question 1: difficult

A uniform pole of length l and mass m is pivoted on the ground with a frictionless hinge O. The pole is free to rotate without friction about an horizontal axis axis passing through O and normal to plane of the page. The pole makes an angle θ with the horizontal. The pole is released from rest in the position shown, then linear acceleration of the free end (P) of the pole just after its release would be

1. (2g cosθ)/3

2. 2g/3

3. g

4. (3g cosθ)/2

View Answer

Torque of gravitational force mg about O is mgcosθ ×L/2

Torque = I × α

mgcosθ ×L/2 = (mL²/3) × α

α = (3gcosθ)/2

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Question 2: moderate

A constant torque of 100 N m turns a wheel of moment of inertia \(300\text{ kg m}^2\) about an axis passing through its centre. Starting from rest, its angular velocity after 3 s is

1. 10 rad/s

2. 15 rad/s

3. 1 rad/s

4. 5 rad/s

View Answer

The angular acceleration is \(\alpha = \frac{\tau}{I} = \frac{100}{300} = \frac{1}{3}\text{ rad/s}^2\). The final angular velocity starting from rest is \(\omega = \alpha t = \frac{1}{3} \times 3 = 1\text{ rad/s}\).

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Question 3: easy

A uniform circular disc of mass \(2\text{ kg}\) and radius \(100\text{ cm}\) (hinged at centre) is subjected to a constant torque of \(4\text{ Nm}\). If the disc was initially at rest, then its angular speed after \(4\text{ s}\) will be

1. 16 rad/s

2. 4 rad/s

3. 2 rad/s

4. 8 rad/s

View Answer

Moment of inertia \(I = \frac{1}{2} M R^2 = \frac{1}{2} \times 2 \times 1^2 = 1\text{ kg m}^2\). Angular acceleration \(\alpha = \frac{\tau}{I} = \frac{4}{1} = 4\text{ rad/s}^2\). Angular speed \(\omega = \alpha t = 4 \times 4 = 16\text{ rad/s}\).

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