Question

Consider a body of mass $1.0\, kg$ 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 $\alpha = 1.0 \, Ns^{-1}$ and $\beta = 1.0 \, N$. The torque acting on the body about the origin at time $t = 1.0 \, s$ is $\vec{\tau}$. Which of the following statements is (are) true?

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