A uniform rod of mass m and length l rotates in a horizontal...

A uniform rod of mass $m$ and length $l$ rotates in a horizontal plane with an angular velocity $\omega$ about a vertical axis passing through one end. The tension in the rod at a distance $x$ from the axis is (a) $\frac{1}{2} m \omega^{2} x$ (b) $\frac{1}{2} m \omega^{2} \frac{x^{2}}{l}$ (c) $\frac{1}{2} m \omega^{2}\left(1-\frac{x}{l}\right)$ (d) $\frac{1}{2} \frac{m \omega^{2}}{l}\left[l^{2}-x^{2}\right]$

A uniform rod of mass $m$ and length $l$ rotates in a horizontal plane with an angular velocity $\omega$ about a vertical axis passing through one end. The tension in the rod at a distance $x$ from the axis is
(a) $\frac{1}{2} m \omega^{2} x$
(b) $\frac{1}{2} m \omega^{2} \frac{x^{2}}{l}$
(c) $\frac{1}{2} m \omega^{2}\left(1-\frac{x}{l}\right)$
(d) $\frac{1}{2} \frac{m \omega^{2}}{l}\left[l^{2}-x^{2}\right]$

Key Concepts

Differential Element Analysis

This technique involves breaking a continuous object into infinitesimally small segments to analyze the forces acting on each segment separately. By applying the principles of dynamics to these differential elements, one can derive equations that describe how physical quantities, such as tension, vary along the length of the body.

Integration in Continuous Systems

Integration is used to sum the contributions from all the differential elements of a continuous object. In problems involving distributed forces, such as the tension in a rotating rod, integration allows one to account for the cumulative effect of the forces acting over the entire length of the object.

Rotational Dynamics

This concept concerns the motion of bodies rotating about an axis. It involves understanding angular velocity, moments of inertia, and how forces act on different parts of the rotating object to maintain the circular motion. It is essential for analyzing any system where components move in a circular path under the influence of rotational motion.

Centripetal Force

Centripetal force is the net inward force required to keep an object moving in a circular path. In the context of rotating rigid bodies, each infinitesimal mass element experiences centripetal acceleration that must be balanced by an inward force, which in this case is provided by the tension in the rod.

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