A uniform horizontal rod of mass m and length ℓ, initially at rest, is free to rotate about a ve

step 1 So in this question we have a horizontal rod of mass m and length l. So suppose this is the rod,

A uniform horizontal rod of mass m and length ℓ, initially...

A uniform horizontal rod of mass $\mathrm{m}$ and length $\ell$, initially at rest, is free to rotate about a vertical axis through its centre. It is subjected to constant horizontal force $\mathrm{F}$ acting on the rod at a distance of $\frac{\ell}{4}$ from the centre and always perpendicular to the rod. The angle of rotation of the rod at the end of time $\mathrm{t}$ after commencement of motion is (a) $\frac{2 \mathrm{Ft}^{2}}{5 \mathrm{~m} \ell}$ (b) $\frac{5 \mathrm{Ft}^{2}}{2 \mathrm{~m} \ell}$ (c) $\frac{3 \mathrm{~F} \ell^{2}}{2 \mathrm{mt}}$ (d) $\frac{3 \mathrm{Ft}^{2}}{2 \mathrm{~m} \ell}$

A uniform horizontal rod of mass $\mathrm{m}$ and length $\ell$, initially at rest, is free to rotate about a vertical axis through its centre. It is subjected to constant horizontal force $\mathrm{F}$ acting on the rod at a distance of $\frac{\ell}{4}$ from the centre and always perpendicular to the rod. The angle of rotation of the rod at the end of time $\mathrm{t}$ after commencement of motion is
(a) $\frac{2 \mathrm{Ft}^{2}}{5 \mathrm{~m} \ell}$
(b) $\frac{5 \mathrm{Ft}^{2}}{2 \mathrm{~m} \ell}$
(c) $\frac{3 \mathrm{~F} \ell^{2}}{2 \mathrm{mt}}$
(d) $\frac{3 \mathrm{Ft}^{2}}{2 \mathrm{~m} \ell}$

Key Concepts

Torque

Torque is the rotational equivalent of force and is responsible for turning an object about an axis. It is calculated as the product of the force applied and the lever arm, which is the perpendicular distance from the axis of rotation to the line of action of the force. In problems involving rotation, understanding how force creates a turning effect is essential for analyzing the motion.

Moment of Inertia

The moment of inertia is a measure of an object's resistance to changes in its rotational motion. It depends on both the mass of the object and how that mass is distributed relative to the axis of rotation. For a rod rotating about its center, this concept helps determine how the applied torque translates into angular acceleration.

Angular Acceleration

Angular acceleration is the rate at which the angular velocity of a rotating object changes over time. It is directly proportional to the applied net torque and inversely proportional to the moment of inertia of the object. In constant torque scenarios, it results in a uniform angular acceleration, similar to linear acceleration in translational motion.

Rotational Kinematics

Rotational kinematics involves the equations that describe the motion of rotating objects under constant angular acceleration. These equations parallel linear kinematics and are used to relate variables such as angular displacement, initial angular velocity, angular acceleration, and time. They are critical for solving problems where the timing and extent of rotation are required.

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