A uniform rod A B of mass m and length l is at rest on a...

A uniform rod $A B$ of mass $m$ and length $l$ is at rest on a smooth horizontal surface. An impulse $p$ is applied to the end $B$. The time taken by the rod to turn through a right angle is : (a) $2 \pi \frac{m l}{p}$ (b) $2 \pi \frac{p}{m l}$ (i) $\frac{\pi m l}{12 p}$ (d) $\frac{\pi p}{m !}$

A uniform rod $A B$ of mass $m$ and length $l$ is at rest on a smooth horizontal surface. An impulse $p$ is applied to the end $B$. The time taken by the rod to turn through a right angle is :
(a) $2 \pi \frac{m l}{p}$
(b) $2 \pi \frac{p}{m l}$
(i) $\frac{\pi m l}{12 p}$
(d) $\frac{\pi p}{m !}$

Key Concepts

Angular Impulse and Angular Momentum

When an impulse is applied at a point away from an object's center of mass, it generates not only a linear momentum but also an angular momentum about the center of mass. This angular impulse, which is the product of the impulse and the perpendicular distance from the center of mass, determines the initial rotational state of the object.

Impulse-Momentum Principle

This concept states that an impulse applied to an object produces a change in its linear momentum. In problems like this, the impulse quantifies the total effect of a force applied over a short time, allowing the translation of momentum to be computed for a system initially at rest.

Moment of Inertia

The moment of inertia is a measure of an object's resistance to angular acceleration about an axis. For rigid bodies, it depends on the mass distribution relative to the axis of rotation. In the case of a uniform rod, using the appropriate moment of inertia is essential for calculating the subsequent angular acceleration or angular speed after the impulse.

Rotational Kinematics

Rotational kinematics involves the study of rotational motion without reference to the forces causing the motion. Using the initial angular velocity determined by the applied impulse and the moment of inertia, one can calculate the time required for the object to rotate through a specific angle, such as a right angle, under constant angular motion or acceleration.

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