Question

Consider a body of mass $1.0kg$ at rest at the origin at time $t=0$. A force $\vec{F}=(\alpha t\hat{i}+\beta \hat{j})$ is applied on the body, where $\alpha =1.0N{s}^{-1}$ and $\beta =1.0N$. The torque acting on the body about the origin at time $t=1.0s$ is $\vec{\tau }$. Which of the following statements is (are) true?

• A. $|\vec{\tau }|=\frac{1}{3}Nm$
• B. The torque $\vec{\tau }$ is in the direction of the unit vector + $\hat{k}$
• C. The velocity of the body at $t=1s$ is $\vec{v}=\frac{1}{2}(\hat{i}+2\hat{j})m{s}^{-1}$
• D. The magnitude of displacement of the body at $t=1s$ is $\frac{1}{6}m$

Answer 1

Answer: (A), (C)

$\vec{F}=\left(\alpha t\right)\hat{i}+\beta \hat{j}$ [At t = 0, v = 0, $\bar{r}=\bar{0}]$
$\alpha =1,\beta =1$
$\vec{F}=t\hat{i}+\hat{j}$
$m\frac{d\vec{v}}{dt}=t\hat{i}+\hat{j}$
On integrating
$m\vec{v}=\frac{t^2}{2}\hat{i}+t\hat{j}$ [m = 1kg]
$\frac{d\vec{r}}{dt}=\frac{t^2}{2}\hat{i}+t\hat{j}$ $[\vec{r}=\vec{0}att=0]$
On integrating
$\vec{r}=\frac{t^3}{6}\hat{i}+\frac{t^2}{2}\hat{j}$
At t = 1 sec, $\vec{\tau }=\left(\vec{r}\times \vec{F}\right)=\left(\frac{1}{6}\hat{i}+\frac{1}{2}\hat{j}\right)\times \left(\hat{i}+\hat{j}\right)$
$\vec{\tau }=-\frac{1}{3}\hat{k}$
$\bar{v}=\frac{t^2}{2}\hat{i}+t\hat{j}$
At $t=1\vec{v}=\left(\frac{1}{2}\hat{i}+\hat{j}\right)=\frac{1}{2}\left(\hat{i}+2\hat{j}\right)m/\sec$
At $t=1\vec{s}=\overrightarrow{r_1}-\overrightarrow{r_0}$
$=\left[\frac{1}{6}\hat{i}+\frac{1}{2}\hat{j}\right]-[\vec{0}]$
$\left|\vec{s}\right|=\sqrt{{\left(\frac{1}{6}\right)}^2+{\left(\frac{1}{2}\right)}^2}\Rightarrow \frac{\sqrt{10}}{6}m$