JEE Advanced 2016 · Paper 1 · Physics Q12

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PhysicsMultiple correct+4 / -2Question 12 of 54

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 $\alpha=10/3\ \mathrm{m\,s^{-3}}$ , $\beta=5\ \mathrm{m\,s^{-2}}$ and $m=0.1$ kg. At $t=1$ s, which of the following statement(s) is(are) true about the particle?

The velocity

\vec v

is given by

\vec v=(10\hat i+10\hat j)\ \mathrm{m\,s^{-1}}

The angular momentum

\vec L

with respect to the origin is given by

\vec L=-(5/3)\hat k

N m s

The force

\vec F

is given by

\vec F=(\hat i+2\hat j)

The torque

\vec\tau

with respect to the origin is given by

\vec\tau=-(20/3)\hat k

N m

Answer and worked solution

Official answer: A, B, D

Kinematics.
$\vec v=3\alpha t^2\hat i+2\beta t\hat j=(10\hat i+10\hat j)\text{ m s}^{-1}\text{ at }t=1.$
$\vec a=6\alpha t\hat i+2\beta\hat j=(20\hat i+10\hat j)\text{ m s}^{-2}.$
The position at $t=1$ is $\vec r=\frac{10}3\hat i+5\hat j$ .

(A) True.
(B) $\vec L=m\,\vec r\times\vec v=0.1\left(\frac{10}3\cdot10-5\cdot10\right)\hat k=-\frac53\hat k$ N m s. True.
(C) $\vec F=m\vec a=(2\hat i+\hat j)$ N. False.
(D) $\vec\tau=\vec r\times\vec F=\left(\frac{10}3\cdot1-5\cdot2\right)\hat k=-\frac{20}3\hat k$ N m. True.

(A), (B), (D)

Topics:Angular momentum and its conservation Motion in a plane Torque and rotational dynamics

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JEE Advanced 2016 · Paper 1, Physics question 12