A uniform rod O A of mass M and length 2 πrests on a smooth table and is free to turn about a sm

step 1 Our question is a uniform rod OA of mass M and length to pi rests on a smooth table and is free

A uniform rod O A of mass M and length 2 πrests on a smooth...

A uniform rod $O A$ of mass $M$ and length $2 \pi$ rests on a smooth table and is free to turn about a smooth pivot at its end $O$, in contact with it at a distance $b$ from $O$ is an inelastic particle of mass $m$, a horizontal blow of impulse $p$ is given to rod at a distance $x$ from $O$ in a direction perpendicular to the rod. The resultant instantaneous angular velocity of the rod is: (a) $\frac{p x}{\frac{4 M a^{2}}{3}+m b^{2}}$ (b) $\frac{p x}{M}$ (c) $\frac{p x}{m a^{2}+m b^{2}}$ (d) none of these

A uniform rod $O A$ of mass $M$ and length $2 \pi$ rests on a smooth table and is free to turn about a smooth pivot at its end $O$, in contact with it at a distance $b$ from $O$ is an inelastic particle of mass $m$, a horizontal blow of impulse $p$ is given to rod at a distance $x$ from $O$ in a direction perpendicular to the rod. The resultant instantaneous angular velocity of the rod is:
(a) $\frac{p x}{\frac{4 M a^{2}}{3}+m b^{2}}$
(b) $\frac{p x}{M}$
(c) $\frac{p x}{m a^{2}+m b^{2}}$
(d) none of these

Key Concepts

Rigid Body Rotation

Rigid body rotation refers to the motion of a body that rotates about a fixed axis where all points of the body maintain fixed distances relative to each other. When external forces or impulses are applied, understanding how these act to change the rotational state of the body through angular momentum conservation and inertia considerations is key to predicting its angular velocity.

Impulse Application in Rigid Body Dynamics

An impulse applied perpendicularly to a rigid body at a point away from its pivot provides an angular impulse. This angular impulse, obtained by multiplying the impulse magnitude by the perpendicular distance from the pivot (the lever arm), directly influences the body’s angular momentum. Analyzing this effect is essential for linking the applied impulse to the resulting rotational motion.

Angular Momentum Conservation

This principle dictates that if no external torque acts on a system, the total angular momentum before an event equals the total angular momentum after the event. In problems involving an impulse applied to a pivoted body, the angular impulse (the product of the impulse magnitude and its lever arm) is equal to the change in angular momentum, allowing for the determination of the body's instantaneous angular velocity.

Moment of Inertia Calculation

The moment of inertia represents the resistance of a body to angular acceleration about a specific axis. For composite systems, such as a rod with an attached particle, the total moment of inertia is the sum of the individual moments of inertia. For a rod pivoted at one end and for a point mass at a known distance from the pivot, standard formulas are used to compute their contributions, which then are combined in determining the rotational dynamics.

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