Question

A solid cone hangs from a frictionless pivot at the origin $O$ as shown. If $\hat{i},\hat{j},\hat{k}$ are unit vectors and $a,b,$ and $c$ are positive constants, determine the torque generated about the origin by a force $F=a\hat{j}\text{N}$ applied to the rim of the cone at a point $P(-b,0,–c)$?

• A. $\overrightarrow{\tau }=ab\hat{i}$
• B. $\overrightarrow{\tau }=-ab\hat{i}$
• C. $\overrightarrow{\tau }=-ab\hat{k}+ac\hat{i}$
• D. None of these

Answer 1

The correct option is C $\overrightarrow{\tau }=-ab\hat{k}+ac\hat{i}$

We know that,
Torque $\overrightarrow{\tau }=\overrightarrow{r}\times \overrightarrow{F}$
For $\overrightarrow{F}=a\hat{j}$ acting on the rim at a point $P(-b,0,-c)$
$\overrightarrow{r}=(-b-0)\hat{i}+(0-0)\hat{j}+(-c-0)\hat{k}=-b\hat{i}-c\hat{k}$
$\therefore \overrightarrow{\tau }=(-b\hat{i}-c\hat{k})\times a\hat{j}=-ab\hat{k}+ac\hat{i}$
Hence, option (c) is correct.