The position vector of a particle of mass is given by the following equation where , and . At , which of the following statement(s) is(are) true about the particle? [2016]

Question

The position vector $\vec{r}$ of a particle of mass $m$ is given by the following equation $\vec{r} (t) = α t^{3} \hat{i} + β t^{2} \hat{j},$

where $α = \frac{10}{3} {ms}^{- 3}$ , $β = 5 {ms}^{- 2}$ and $m = 0.1 kg$ . At $t = 1 s$ , which of the following statement(s) is(are) true about the particle? [2016]

Smart Achievers · Accepted Answer

(1, 2, 4)

$Given : \vec{r} = α t^{3} \hat{i} + β t^{2} \hat{j}$

$\vec{r} = \frac{10}{3} t^{3} \hat{i} + 5 t^{2} \hat{j} m$

$∴$ $\vec{v} = \frac{d \vec{r}}{d t} = 10 t^{2} \hat{i} + 10 t \hat{j} {ms}^{- 1}$

$and \vec{a} = \frac{d \vec{v}}{d t} = 20 t \hat{i} + 10 \hat{j} {ms}^{- 2}$

$At t = 1 s, {\vec{r}}_{t = 1} = \frac{10}{3} \hat{i} + 5 \hat{j} m$

${\vec{v}}_{t = 1} = 10 \hat{i} + 10 \hat{j} {ms}^{- 1}, {\vec{a}}_{t = 1} = 20 \hat{i} + 10 \hat{j} {ms}^{- 2}$

${\vec{p}}_{t = 1} = \hat{i} + \hat{j} {kg ms}^{- 1}$

$\vec{L} = \vec{r} \times \vec{p} =$ $| \begin{matrix} \hat{i} & \hat{j} & \hat{k} \\ \frac{10}{3} & 5 & 0 \\ 1 & 1 & 0 \end{matrix} |$ $= \hat{k} [\frac{10}{3} - 5] = - \frac{5}{3} \hat{k} {kg ms}^{- 1}$

$\vec{F} = m \vec{a} = (2 \hat{i} + \hat{j}) N$

$\vec{τ} = \vec{r} \times \vec{F} =$ $| \begin{matrix} \hat{i} & \hat{j} & \hat{k} \\ \frac{10}{3} & 5 & 0 \\ 2 & 1 & 0 \end{matrix} |$ $= \hat{k} [\frac{10}{3} - 10] = - \frac{20}{3} \hat{k} Nm$

Solution

1 The velocity $\vec{v}$ is given by $\vec{v} = (10 \hat{i} + 10 \hat{j}) {ms}^{- 1}$

2 The angular momentum $\vec{L}$ with respect to the origin is given by $\vec{L} = - \frac{5}{3} \hat{k} N m s$

3 The force $\vec{F}$ is given by $\vec{F} = (\hat{i} + 2 \hat{j}) N$

4 The torque $\vec{τ}$ with respect to the origin is given by $\vec{τ} = - \frac{20}{3} \hat{k} N m$

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Solution

Q. The position vector r→ of a particle of mass m is given by the following equation r→(t)=αt3i^+βt2j^, where α=103ms-3, β=5 ms-2 and m=0.1 kg. At t=1 s, which of the following statement(s) is(are) true about the particle [2016]

1 The velocity v→ is given by v→=(10i^+10j^) ms-1

2 The angular momentum L→ with respect to the origin is given by L→=-53k^ N m s

3 The force F→ is given by F→=(i^+2j^) N

4 The torque τ→ with respect to the origin is given by τ→=-203k^ N m

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1 The velocity $\vec{v}$ is given by $\vec{v} = (10 \hat{i} + 10 \hat{j}) {ms}^{- 1}$

2 The angular momentum $\vec{L}$ with respect to the origin is given by $\vec{L} = - \frac{5}{3} \hat{k} N m s$

3 The force $\vec{F}$ is given by $\vec{F} = (\hat{i} + 2 \hat{j}) N$

4 The torque $\vec{τ}$ with respect to the origin is given by $\vec{τ} = - \frac{20}{3} \hat{k} N m$