The position of an object having mass 0.1 kg as a function of time $t$ is given as $\vec{r}=\left(10 t^{2} \hat{i}+5 t^{3} \hat{j}\right) \mathrm{m}$. At $\mathrm{t}=1 \mathrm{~s}$, which of the following statements are correct? A. The linear momentum $\overrightarrow{\mathrm{p}}=(2 \hat{\mathrm{i}}+1.5 \hat{\mathrm{j}}) \mathrm{kg} \cdot \mathrm{m} / \mathrm{s}$. B. The force acting on the object $\vec{F}=(2 \hat{i}+3 \hat{j}) N$. C. The angular momentum of the object about its origin $\overrightarrow{\mathrm{L}}=15 \hat{\mathrm{k}} \mathrm{Js}$. D. The torque acting on the object about its origin $\vec{\tau}=20 \hat{\mathrm{k}} \mathrm{Nm}$. Choose the correct answer from the option given below.

JEE Main · 2026 · 02 Apr 2026 · Shift 1 · Rotational Mechanics

The position of an object having mass 0.1 kg as a function of time

t

is given as

r = (10 t^{2} \hat{i} + 5 t^{3} \hat{j}) m

. At

t = 1 s

, which of the following statements are correct?
A. The linear momentum

p = (2 \hat{i} + 1.5 \hat{j}) kg \cdot m / s

.
B. The force acting on the object

F = (2 \hat{i} + 3 \hat{j}) N

.
C. The angular momentum of the object about its origin

L = 15 \hat{k} Js

.
D. The torque acting on the object about its origin

τ = 20 \hat{k} Nm

.

Choose the correct answer from the option given below.

Answer: (D)

t = 1

v = \frac{d r}{d t} = 20 t \hat{i} + 15 t^{2} \hat{j} = 20 \hat{i} + 15 \hat{j}

a = 20 \hat{i} + 30 t \hat{j} = 20 \hat{i} + 30 \hat{j}

, and

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

. The momentum

p = m v = 0.1 (20 \hat{i} + 15 \hat{j}) = 2 \hat{i} + 1.5 \hat{j}

confirms statement A, and the force

F = m a = 0.1 (20 \hat{i} + 30 \hat{j}) = 2 \hat{i} + 3 \hat{j}

confirms statement B. The torque

τ = r \times F = (10 \hat{i} + 5 \hat{j}) \times (2 \hat{i} + 3 \hat{j}) = (30 - 10) \hat{k} = 20 \hat{k}

N\,m confirms statement D. The angular momentum

L = r \times p = (10 \hat{i} + 5 \hat{j}) \times (2 \hat{i} + 1.5 \hat{j}) = (15 - 10) \hat{k} = 5 \hat{k}

J\,s, not

15 \hat{k}

, so statement C is wrong. Statements A, B and D are correct.

JEE Main · 2026 · 02 Apr 2026 · Shift 1 · Single correct option