Consider a body of mass $1.0 kg$ at rest at the origin at time $t=0$. A force $\vec{F}=(\alpha t \hat{\imath}+\beta \hat{\jmath})$ is applied on the body, where $\alpha=1.0 N s^{-1}$ 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) $|\vec{\tau}|=\frac{1}{3} N m$; (C) The velocity of the body at $t=1 s$ is $\overrightarrow{\mathrm{v}}=\frac{1}{2}(\hat{\imath}+2 \hat{\jmath}) m s^{-1}$

(B) The torque $\vec{\tau}$ is in the direction of the unit vector $+\widehat{k}$

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

JEE Advanced · 2018 · 20 May 2018 · Paper 1 · Rotational Mechanics

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Consider a body of mass

1.0 k g

at rest at the origin at time

t = 0

. A force

F = (α t \overset{}{^} + β \overset{}{^})

is applied on the body, where

α = 1.0 N s^{- 1}

and

β = 1.0 N

. The torque acting on the body about the origin at time

t = 1.0 s

τ

. Which of the following statements is (are) true?

(A) $∣ τ ∣ = \frac{1}{3} N m$ ✓ Correct
(B)The torque $τ$ is in the direction of the unit vector $+ k$
(C)The velocity of the body at $t = 1 s$ is $v = \frac{1}{2} (\overset{}{^} + 2 \overset{}{^}) m s^{- 1}$ ✓ Correct
(D)The magnitude of displacement of the body at $t = 1 s$ is $\frac{1}{6} m$

Answer: (A), (C)

Solution

The acceleration is

a = F / m = t \hat{i} + \hat{j}

. Integrating:

v = \int_{0}^{t} (t \hat{i} + \hat{j}) d t = \frac{t ^{2}}{2} \hat{i} + t \hat{j}

. At

t = 1 s

v = \frac{1}{2} \hat{i} + \hat{j} = \frac{1}{2} (\hat{i} + 2 \hat{j})

m/s. Position:

r = \frac{t ^{3}}{6} \hat{i} + \frac{t ^{2}}{2} \hat{j}

. At

t = 1 s

r = \frac{1}{6} \hat{i} + \frac{1}{2} \hat{j}

. Torque:

τ = r \times F = (\frac{1}{6} \hat{i} + \frac{1}{2} \hat{j}) \times (\hat{i} + \hat{j}) = - \frac{1}{3} \hat{k}

. Thus

∣ τ ∣ = \frac{1}{3}

Nm and it's in the

- \hat{k}

direction.

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