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

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$|τ| = \frac{1}{3} N . m$

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

The torque 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.

answer is A, C.

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Detailed Solution

$\begin{array}{l} F = t \hat{i} + \hat{j} \\ m \frac{d \vec{v}}{d t} = t \hat{i} + \hat{j} \\ m \vec{v} = \frac{t^{2}}{2} \hat{i} + t \hat{j} \\ \frac{d \vec{r}}{d t} = \frac{t^{2}}{2} \hat{i} + t \hat{j} \\ \vec{r} = \frac{t^{3}}{6} \hat{i} + \frac{t^{2}}{2} \hat{j} \end{array}$

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