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A body of mass 1kg at rest at the origin when t = 0. A force $F = (αt \hat{i} + βj)$ is applied on the body. $α = 1 N s^{- 1}; β = 1 N$ . The Torque acted on the body about the origin at t =1sec is $\bar{τ}$ . Which of the following is (are) True?

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$|τ| = \frac{1}{3} N . m$

Toque is $τ$ k

Velocity after 1sec ; $\bar{ν} = \frac{1}{2} (\hat{i} + 2 j) m / s$

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

answer is A, C.

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Detailed Solution

$F = t \hat{i} + \hat{j} = \frac{\vec{F}}{m} = (t \hat{i} + j)$

$\frac{d ν}{dt} = t \hat{i} + j \Rightarrow \vec{ν} = \frac{t^{2} \hat{i}}{2} + t \hat{j}$

$\Rightarrow if t=1; \vec{ϑ} = \frac{1}{2} (i + 2 j) m/s$

$\frac{\vec{d r}}{dt} = \frac{t^{2} \hat{i}}{2} + t j$

$\Rightarrow \vec{r} = \frac{t^{2}}{6} \hat{i} + \frac{t^{2}}{2} j$

$\vec{τ} = \vec{r} \times \vec{F} = - \frac{1}{3} t^{3} k t = 1; \vec{τ = - \frac{1}{3}} kNm$

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