Question

Let $F$ be the force acting on a particle having position vector $r$ and $\tau $ be the torque of this force about the origin. Then,
A. $r\cdot \tau =0$ and $F\cdot \tau \ne 0$
B. $r\cdot \tau \ne 0$ and $F\cdot \tau =0$
C. $r\cdot \tau \ne 0$ and $F\cdot \tau \ne 0$
D. $r\cdot \tau =0$ and $F\cdot \tau =0$

Answer 1

Hint: In order to solve this question we will be learning what is meant by the terms force and torque. Torque is the measure of the force that can cause an object to rotate about an axis. Force is what causes an object to accelerate in linear kinematics.Similarly, torque is what causes an angular acceleration. Hence, torque can be defined as the rotational equivalent of linear force.

Complete step by step answer:
First we will know what force is. Force is any interaction that can change the motion of an object. It can cause any object with mass to change its motion. Force can also be described as push or a pull. Force is a vector quantity (i.e. it has both magnitude as well as direction). SI unit of force is newton $\left( N \right)$.

Now we will know about Torque. Another name of torque is moment of force. Torque or moment of force is the tendency of a force to rotate or orient an object about an axis, fulcrum, or pivot. Just as force is a push or a pull, a torque can be thought of as a twist to any object. Torque is denoted by $\tau $. Torque is also a vector quantity and its SI units are newton meters $\left( Nm \right)$. Torque is the cross product of the force and the position vector $\overrightarrow{\tau }=\overrightarrow{r}\times \overrightarrow{F}$. Here torque is acting about the origin so the angle between $\overrightarrow{r}$ and $\overrightarrow{\tau }$ is ${{90}^{{}^\circ }}$ and between $\overrightarrow{F}$ and $\overrightarrow{\tau }$ is also ${{90}^{{}^\circ }}$.

Now when we take dot product of two vectors $\overrightarrow{r}$ and $\overrightarrow{\tau }$ as $\theta ={{90}^{{}^\circ }}$ we get
$\overrightarrow{r}\cdot \overrightarrow{\tau }= r\tau \cos {{90}^{\circ }}$
$\Rightarrow \overrightarrow{r}\cdot \overrightarrow{\tau }=0$
$\therefore r\cdot \tau =0$
Similarly when we take dot product of$\overrightarrow{F}$ and $\overrightarrow{\tau }$, as $\theta ={{90}^{{}^\circ }}$ we get
$\overrightarrow{F}\cdot \overrightarrow{\tau }=F\tau \cos {{90}^{\circ }}$
$\Rightarrow \overrightarrow{F}\cdot \overrightarrow{\tau }=0$
$\therefore F\cdot \tau =0$

Hence, the correct answer is option D.

Note: Torque due to a force is more if the distance between the point of application of force and the point about which the body rotates is more. If the distance at which force is applied is more from the fulcrum then a lesser amount of force is required to produce the same torque.