Question

If $\vec{r} = 10t^2 \hat{i} + 5t^3 \hat{j}$ and mass of object, $m = 0.1 \text{ kg}$ then at $t = 1 \text{ sec}$ :-
(A) momentum = $2\hat{i} + 1.5\hat{j}$
(B) force = $2\hat{i} + 3\hat{j}$
(C) Angular momentum = $5\hat{k}$
(D) Torque = $20\hat{k}$

• A. A, B, C are correct
• B. A, C, D are correct
• C. A, C are correct
• D. A, B, C, D are correct

Answer 1

The Correct Option is D

Step 1: Find velocity $\vec{v}$ and acceleration $\vec{a}$ by differentiating the position vector.
Position: $\vec{r} = 10t^2 \hat{i} + 5t^3 \hat{j}$
Velocity: $\vec{v} = \frac{d\vec{r}}{dt} = 20t \hat{i} + 15t^2 \hat{j}$
Acceleration: $\vec{a} = \frac{d\vec{v}}{dt} = 20 \hat{i} + 30t \hat{j}$

Step 2: Calculate values at $t = 1 \text{ s}$.
$\vec{r} = 10\hat{i} + 5\hat{j}$
$\vec{v} = 20\hat{i} + 15\hat{j}$
$\vec{a} = 20\hat{i} + 30\hat{j}$

Step 3: Evaluate statement (A) Momentum ($\vec{P}$).
$\vec{P} = m\vec{v} = 0.1(20\hat{i} + 15\hat{j}) = 2\hat{i} + 1.5\hat{j}$. (A is correct)

Step 4: Evaluate statement (B) Force ($\vec{F}$).
$\vec{F} = m\vec{a} = 0.1(20\hat{i} + 30\hat{j}) = 2\hat{i} + 3\hat{j}$. (B is correct)

Step 5: Evaluate statement (C) Angular momentum ($\vec{L}$).
$\vec{L} = \vec{r} \times \vec{P} = (10\hat{i} + 5\hat{j}) \times (2\hat{i} + 1.5\hat{j})$
$\vec{L} = [(10 \times 1.5) - (5 \times 2)] \hat{k} = (15 - 10)\hat{k} = 5\hat{k}$. (C is correct)

Step 6: Evaluate statement (D) Torque ($\vec{\tau}$).
$\vec{\tau} = \vec{r} \times \vec{F} = (10\hat{i} + 5\hat{j}) \times (2\hat{i} + 3\hat{j})$
$\vec{\tau} = [(10 \times 3) - (5 \times 2)] \hat{k} = (30 - 10)\hat{k} = 20\hat{k}$. (D is correct)

All statements are correct. Final Answer: Option (4).