A rod of length L is pivoted at one end and is rotated with...

A rod of length $L$ is pivoted at one end and is rotated with a uniform angular velocity in a horizontal plane. Let $T_{1}$ and $T_{2}$ be the tensions at the points $\frac{L}{4}$ and $\frac{3 L}{4}$ away from the pivoted end. Then: (a) $T_{1}>T_{2}$ (b) $T_{2}>T_{1}$ (c) $T_{1}=T_{2}$ (d) the relation between $T_{1}$ and $T_{2}$ depends on whether the rod rotates clockwise or anti-clockwise

A rod of length $L$ is pivoted at one end and is rotated with a uniform angular velocity in a horizontal plane. Let $T_{1}$ and $T_{2}$ be the tensions at the points $\frac{L}{4}$ and $\frac{3 L}{4}$ away from the pivoted end. Then:
(a) $T_{1}>T_{2}$
(b) $T_{2}>T_{1}$
(c) $T_{1}=T_{2}$
(d) the relation between $T_{1}$ and $T_{2}$ depends on whether the rod rotates clockwise or anti-clockwise

Key Concepts

Rotational Motion

Rotational motion involves the movement of bodies around a fixed axis or pivot. In the context of a rigid body rotating with uniform angular velocity, every point on the object moves in a circular path. This concept is crucial because it helps to determine the dynamics associated with the rotation, such as angular acceleration, angular momentum, and the distribution of kinematic requirements like the centripetal acceleration that each segment of the object must experience.

Centripetal Force

Centripetal force is the inward force required to keep an object moving in a circular path. For a rotating body, every mass element must be provided with the necessary centripetal force to maintain its circular motion, which is proportional to the square of the angular velocity and its distance from the axis of rotation. This concept is essential for understanding how force requirements vary along a rotating object.

Tension Distribution in Rigid Bodies

Tension distribution in a rotating rigid body relates to how internal forces, such as tension, vary along its length due to the need to supply centripetal forces to successive segments of the body. Points closer to the pivot typically bear more tension because they must support the centripetal force requirements of the entire length of the rod beyond them, while points farther out support a lesser portion of the overall required force. This concept helps in understanding the gradation of internal forces along a continuously distributed mass.

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