A particle of mass is initially at rest in the -plane at a point , where and . The particle is accelerated at time with a constant acceleration along the positive -direction. Its angular momentum and torque with respect to the origin, in SI units, are

Question

A particle of mass $M = 0.2 kg$ is initially at rest in the $x y$ -plane at a point $(x = - l, y = - h)$ , where $l = 10 m$ and $h = 1 m$ . The particle is accelerated at time $t = 0$ with a constant acceleration $a = 10 {m/s}^{2}$ along the positive $x$ -direction. Its angular momentum and torque with respect to the origin, in SI units, are represented by $\vec{L}$ and $\vec{τ}$ , respectively. $\hat{i}, \hat{j}$ and $\hat{k}$ are unit vectors along the positive $x$ , $y$ and $z$ -directions, respectively. If $\hat{k} = \hat{i} \times \hat{j}$ then which of the following statement(s) is(are) correct? [2021]

Smart Achievers · Accepted Answer

(1, 2, 3)

Particle is initially at rest in the $x y$ -plane at a point $x = - ℓ, y = - h$ where $ℓ = 10$ m and $h = 1 m$

$From s = u t + \frac{1}{2} a t^{2} \Rightarrow 20 = \frac{1}{2} \times 10 \times t^{2}$

$∴$ $\hat{i}, \hat{j}$

$\hat{k}$

$\vec{F} = m \vec{a} = 0.2 \times 10 \hat{i} = 2 \hat{i}$ $∴$ $\vec{τ} = (10 \hat{i} - \hat{j}) \times (2 \hat{i}) = 2 \hat{k}$

$Angular momentum, \vec{L} = {\vec{r}}_{B} \times \vec{P} = {\vec{r}}_{B} \times m \vec{v}$

$From \vec{v} = \vec{u} + \vec{a} t = 10 \hat{i} \times 2 = 20 \hat{i}$

$\vec{L} = (0.2) [(10 \hat{i} - \hat{j}) \times 20 \hat{i}] = 4 \hat{k}$

$At point A (0, - 1)$

$\vec{τ} = {\vec{r}}_{A} \times \vec{F} = (- \hat{j}) \times 2 \hat{i} = 2 \hat{k} [∵ {\vec{r}}_{A} = - \hat{j}]$

Solution

1 The particle arrives at the point $(x = l, y = - h)$ at time $t = 2 s$ .

2 $\vec{τ} = 2 \hat{k}$ when the particle passes through the point $(x = l, y = - h)$

3 $\hat{L} = 4 \hat{k}$ when the particle passes through the point $(x = l, y = - h)$

4 $\vec{τ} = \hat{k}$ when the particle passes through the point $(x = 0, y = - h)$

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Solution

1 The particle arrives at the point (x=l,y=-h) at time t=2 s.

2 τ→=2k^ when the particle passes through the point (x=l,y=-h)

3 L^=4k^ when the particle passes through the point (x=l,y=-h)

4 τ→=k^ when the particle passes through the point (x=0,y=-h)

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1 The particle arrives at the point $(x = l, y = - h)$ at time $t = 2 s$ .

2 $\vec{τ} = 2 \hat{k}$ when the particle passes through the point $(x = l, y = - h)$

3 $\hat{L} = 4 \hat{k}$ when the particle passes through the point $(x = l, y = - h)$

4 $\vec{τ} = \hat{k}$ when the particle passes through the point $(x = 0, y = - h)$