Question

Suppose that a rigid body is free to rotate about a fixed point $O$. If a force $\overrightarrow{F}$ acts on the body at a point $P$, then the body will tend to rotate about an axis through $O$. This effect is measured by the torque vector $t$ which is given by $\overrightarrow{t}=\overrightarrow{r}\times \overrightarrow{F}$. Where $\overrightarrow{r}$ is the position vector $\overrightarrow{OP}$. The straight line through $O$ determined by $\overrightarrow{t}$ is the axis of rotation. The vector $\overrightarrow{r},\overrightarrow{F}$ and $\overrightarrow{T}$ form a right handed system.
A bolt is being tightened by a $20N$ force is applied to a $25cm$ long wrench at an angle of ${60}^0$ with the wrench. The magnitude of torque is

• A. $100\sqrt{3}$
• B. $500\sqrt{3}$
• C. $250\sqrt{3}$
• D. $350\sqrt{3}$

Answer 1

The correct option is C $250\sqrt{3}$


$\overrightarrow{r}=25\hat{i}$
$\overrightarrow{F}=20\cos {60}^0\hat{i}+20\sin {60}^0\hat{i}$
$\overrightarrow{F}=10\hat{i}+10\sqrt{3}\hat{j}$
$\overrightarrow{t}=\overrightarrow{r}\times \overrightarrow{F}$
$\overrightarrow{t}=\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ 25& 0& 0\\ 10& 10\sqrt{3}& 0\end{array}\right|=(0-0)\hat{i}-(0-0)\hat{j}+(250\sqrt{3}-0)\hat{k}$
$\overrightarrow{t}=250\sqrt{3}\hat{k}$
$|\overrightarrow{t}|=\sqrt{(250\sqrt{3}{)}^2}$
$|\overrightarrow{t}|=250\sqrt{3}$