The position vector $\vec{r}$ of a particle of mass $m$ is given by the following equation \[ \vec{r}(t)=\alpha t^{3} \hat{i}+\beta t^{2} \hat{j}, \] where $\alpha=10/3 \mathrm{~m} \mathrm{~s}^{-3}$, $\beta=5 \mathrm{~m} \mathrm{~s}^{-2}$ and $m=0.1 \mathrm{~kg}$. At $t=1 \mathrm{~s}$, which of the following statement(s) is(are) true about the particle? (A) The velocity $\vec{v}$ is given by $\vec{v}=(10 \hat{i}+10 \hat{j}) \mathrm{m s}^{-1}$ (B) The angular momentum $\vec{L}$ with respect to the origin is given by $\vec{L}=-(5/3) \hat{k} \mathrm{~N} \mathrm{~m} \mathrm{~s}$ (C) The force $\vec{F}$ is given by $\vec{F}=(\hat{i}+2 \hat{j}) \mathrm{N}$ (D) The torque $\vec{\tau}$ with respect to the origin is given by $\vec{\tau}=-(20/3) \hat{k} \mathrm{~N} \mathrm{~m}$

(A) The velocity $\vec{v}$ is given by $\vec{v}=(10 \hat{i}+10 \hat{j}) \mathrm{m s}^{-1}$; (B) The angular momentum $\vec{L}$ with respect to the origin is given by $\vec{L}=-(5/3) \hat{k} \mathrm{~N} \mathrm{~m} \mathrm{~s}$; (D) The torque $\vec{\tau}$ with respect to the origin is given by $\vec{\tau}=-(20/3) \hat{k} \mathrm{~N} \mathrm{~m}$

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

JEE Advanced · 2016 · 22 May 2016 · Paper 1 · Rotational Mechanics

WhatsApp Email

The position vector

r

of a particle of mass

m

is given by the following equation

r (t) = α t^{3} \hat{i} + β t^{2} \hat{j},

where

α = 10/3 m s^{- 3}

β = 5 m s^{- 2}

and

m = 0.1 kg

. At

t = 1 s

, which of the following statement(s) is(are) true about the particle?
(A) The velocity

v

is given by

v = (10 \hat{i} + 10 \hat{j}) ms^{- 1}

(B) The angular momentum

L

with respect to the origin is given by

L = - (5/3) \hat{k} N m s

F

is given by

F = (\hat{i} + 2 \hat{j}) N

(D) The torque

τ

with respect to the origin is given by

τ = - (20/3) \hat{k} N m

(A)The velocity $v$ is given by $v = (10 \hat{i} + 10 \hat{j}) ms^{- 1}$ ✓ Correct
(B)The angular momentum $L$ with respect to the origin is given by $L = - (5/3) \hat{k} N m s$ ✓ Correct
(C)The force $F$ is given by $F = (\hat{i} + 2 \hat{j}) N$
(D)The torque $τ$ with respect to the origin is given by $τ = - (20/3) \hat{k} N m$ ✓ Correct

Answer: (A), (B), (D)

Solution

Velocity:

v = \frac{d r}{d t} = 3 α t^{2} \hat{i} + 2 β t \hat{j}

t = 1

v = 3 \times \frac{10}{3} \hat{i} + 2 \times 5 \hat{j} = (10 \hat{i} + 10 \hat{j})

m/s. (A) is correct.

Acceleration:

a = \frac{d v}{d t} = 6 α t \hat{i} + 2 β \hat{j}

t = 1

a = 20 \hat{i} + 10 \hat{j}

m/s²

Position at t=1 s:

r = \frac{10}{3} \hat{i} + 5 \hat{j}

m

Angular momentum:

L = r \times m v = (\frac{10}{3} \hat{i} + 5 \hat{j}) \times 0.1 (10 \hat{i} + 10 \hat{j}) = 0.1 (\frac{10}{3} \times 10 - 5 \times 10) \hat{k} = - \frac{5}{3} \hat{k}

N·m·s. (B) is correct.

Force:

F = m a = 0.1 (20 \hat{i} + 10 \hat{j}) = (2 \hat{i} + \hat{j})

N. (C) is incorrect.

Torque:

τ = r \times F = (\frac{10}{3} \hat{i} + 5 \hat{j}) \times (2 \hat{i} + \hat{j}) = (\frac{10}{3} - 10) \hat{k} = - \frac{20}{3} \hat{k}

N·m. (D) is correct.

JEE Advanced · 2016 · 22 May 2016 · Paper 1 · One or more correct options

Practise more: all JEE Advanced PYQs on Rotational Mechanics →

JEE Advanced · 2016 · 22 May 2016 · Paper 1 · Rotational Mechanics

Solution

More JEE Advanced questions on Rotational Mechanics