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

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$|\vec{τ}| = \frac{1}{3} N m$

The torque $\vec{τ}$ is in the direction of the unit vector $+ \hat{k}$

The velocity of the body at $t = 1 s$ is $\vec{v} = \frac{1}{2} (\hat{i} + 2 \hat{j}) m / s$

The magnitude of displacement of the body at $t = 1 s$ is $\frac{1}{6} m$

answer is A, C.

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Detailed Solution

$\begin{array}{l} \vec{F} = (α t) \hat{i} + β \hat{j} \\ \Rightarrow \vec{F} = t \hat{i} + \hat{j}, \\ \Rightarrow m \frac{d \vec{v}}{d t} = t \hat{i} + \hat{j} \end{array}$

On Integrating

$\begin{array}{l} \Rightarrow m \vec{v} = \frac{t^{2}}{2} \hat{i} + t \hat{j} \\ \Rightarrow \frac{d \vec{r}}{d t} = \frac{t^{2}}{2} \hat{i} + t \hat{j} \end{array}$

On Integrating

$\begin{array}{l} \Rightarrow \vec{r} = \frac{t^{3}}{6} \hat{i} + \frac{t^{2}}{2} \hat{j} \\ At t = 1 s, \vec{r} = (\vec{r} \times \vec{F}) = (\frac{1}{6} \hat{i} + \frac{1}{2} \hat{j}) \times (\hat{i} \times \hat{j}) = - \frac{1}{3} \hat{k} \\ At t = 1 s, \vec{v} = (\frac{1}{2} \hat{i} + \hat{j}) = \frac{1}{2} (\hat{i} + 2 \hat{j}) m / \sec \\ At t = 1 s, \vec{s} = {\vec{r}}_{1} - {\vec{r}}_{0} \\ \Rightarrow \vec{s} = \frac{1}{6} \hat{i} + \frac{1}{2} \hat{j} \\ \Rightarrow | \vec{s} | = \sqrt{{(\frac{1}{6})}^{2} + {(\frac{1}{2})}^{2}} = \frac{\sqrt{10}}{6} m \end{array}$