Question

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 tension at the points $L/4$ and $3L/4$ always from the pivoted end. Then:

• A. $T_1>T_2$
• B. $T_1=T_2$
• C. $T_1<T_2$
• D. The relation between $T_1$ and $T_2$ depends on whether the rod rotates clockwise or anticlockwise.

Answer 1

Answer: (A)

$T_1>T_2$

Let mass per unit length of the rod is $m$

And take a strip of length $dx$ at a distance $x$ from the pivoted end

Tension in the rod is due to the centrifugal force in the part of the rod that is away from the point of interest

So, for the point at a distance $\frac{L}{4}$

$T_1={\int }_{L/4}^L\left(mdx\right)w^{2}xdx$

$=\frac{15}{32}m{w}^2{L}^2$

For a point at $\frac{3L}{4}$

$T_2={\int }_{3L/4}^L\left(mdx\right)w^{2}xdx$

$=\frac{7}{32}m{w}^2{L}^2$

So, $T_1>T_2$