Question

If a body is rotating about a given Axis and a Force is applied at any point on that body along any random direction, what can we say about the torque of this force about any point lying on its Axis of rotation?

• A. The torque will be different about different points along the axis of rotation.
• B. The torque will be same about different point along the axis of rotation
• C. The magnitude of torque will remain the same about different points but the direction would vary.
• D. It depends upon the orientation of the body and therefore not predictable

Answer 1

The correct option is B

The torque will be same about different point along the axis of rotation

Consider a body as shown in figure which can rotate freely about an axis 0 0^I. Then the torque of force $\overrightarrow{T}$
applied at any point P on the body with position

= $\overrightarrow{T}$ = $\overrightarrow{r}$ $x$ $\overrightarrow{F}$

Its projection along OO' {Let say $\text{/}Z$ axis}
= ($\overrightarrow{r}$ $x$ $\overrightarrow{F}$) . $\overrightarrow{k}$

Which is equal if calculated about any point lying on the axis of rotation. This is called torque about O along OO^I

Eg:- AA' is an axis about which the body is free to rotate. Force F is into the plane. The perpendicular distance of F from axis is 4 units. So torque about line AA' will be r x F = 40

Same situation but this time I am choosing a different point on the axis such that its distance from the point of application of force is 5 units.

F is in the plane of paper. $\overrightarrow{T}$ will be perpendicular to the line joining OP. By geometry you can find the angle that Torque makes will axis (37⁰ here).

Now this torque along axis will be
$\overrightarrow{T}$ cos ${37}^{\circ }=5\times 10\times \frac{4}{5}$

Which is same as previous. You can also generalize this by taking any other $\theta$.