Consider a body of mass 1.0 kg at rest at the origin at time . A force is applied on the body, where and . The torque acting on the body about the origin at time is . Which of the following statements is (are) true? [2018]

Question

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 $α = 1.0 {Ns}^{- 1}$ 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? [2018]

Smart Achievers · Accepted Answer

(1, 3)

$Given \vec{F} = α t \hat{i} + β \hat{j} or \vec{F} = t \hat{i} + \hat{j}$ $(∵ α = 1 {Ns}^{- 1}, β = 1 N)$

$∴$ $\frac{m d \vec{v}}{d t} = t \hat{i} + \hat{j}$

$∴$ $d \vec{v} = t d t \hat{i} + d t \hat{j}$ $[∵ m = 1]$

$∴$ $\int_{0}^{v} d \vec{v} = \int_{0}^{t} t d t \hat{i} + \int_{0}^{t} d t \hat{j} \Rightarrow \vec{v} = \frac{t^{2}}{2} \hat{i} + t \hat{j}$

$At t = 1 s, \vec{v} = \frac{1}{2} \hat{i} + \hat{j} = \frac{1}{2} (\hat{i} + 2 \hat{j}) {ms}^{- 1}$

$Also, \vec{v} = \frac{d \vec{r}}{d t} = \frac{t^{2}}{2} \hat{i} + t \hat{j}$ $∴$ $d \vec{r} = \frac{t^{2}}{2} d t \hat{i} + t d t \hat{j}$

$or \int_{0}^{r} d \vec{r} = \int_{0}^{t} \frac{t^{2}}{2} d t \hat{i} + \int_{0}^{t} t d t \hat{j} \Rightarrow \vec{r} = \frac{t^{3}}{6} \hat{i} + \frac{t^{2}}{2} \hat{j}$

$At t = 1, \vec{r} = \frac{1}{6} \hat{i} + \frac{1}{2} \hat{j},$ $∴$ $| \vec{r} |$ $= \sqrt{\frac{1}{36} + \frac{1}{4}} = \sqrt{\frac{10}{36}}$

$\vec{τ} = \vec{r} \times \vec{F} = (\frac{1}{6} \hat{i} + \frac{1}{2} \hat{j}) \times (\hat{i} + \hat{j})$ $(at t = 1 s)$

$or \vec{τ} = - \frac{1}{3} \hat{k}$ $∴$ $| \vec{τ} |$ $= \frac{1}{3} Nm$

Solution

1 $| \vec{τ} |$ $= \frac{1}{3} Nm$

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

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

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

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Solution

Q. Consider a body of mass 1.0 kg at rest at the origin at time t=0. A force F→=(αti^+βj^) 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=1.0 s is τ→. Which of the following statements is (are) true [2018]

1 |τ→|=13 Nm

2 The torque τ→ is in the direction of the unit vector +k^

3 The velocity of the body at t=1 s is v→=12(i^+2j^) ms-1

4 The magnitude of displacement of the body at t=1 s is 16 m

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1 $| \vec{τ} |$ $= \frac{1}{3} Nm$

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

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

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