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

To solve this problem, let's break down each part step-by-step:

1. Objective: We need to determine the momentum, force, angular momentum, and torque of an object whose position vector is given by \(\vec{r} = 10t^2 \hat{i} + 5t^3 \hat{j}\) with mass \(m = 0.1 \text{ kg}\), at \(t = 1 \text{ sec}\).
2. Calculate Momentum:
   • Velocity \(\vec{v}\) is the derivative of \(\vec{r}\) with respect to time \(t\): \(\vec{v} = \frac{d\vec{r}}{dt} = \frac{d}{dt}(10t^2 \hat{i} + 5t^3 \hat{j}) = 20t \hat{i} + 15t^2 \hat{j}\)
   • At \(t = 1\), \(\vec{v} = 20 \times 1 \hat{i} + 15 \times 1^2 \hat{j} = 20 \hat{i} + 15 \hat{j}\)
   • Momentum \(\vec{p} = m \cdot \vec{v} = 0.1 \times (20 \hat{i} + 15 \hat{j}) = 2 \hat{i} + 1.5 \hat{j}\)
3. Calculate Force:
   • Force \(\vec{F}\) is the derivative of momentum \(\vec{p}\) with respect to time \(t\) which is equivalent to mass times acceleration.
   • Acceleration \(\vec{a}\) is the derivative of velocity \(\vec{v}\): \(\vec{a} = \frac{d\vec{v}}{dt} = \frac{d}{dt}(20t \hat{i} + 15t^2 \hat{j}) = 20 \hat{i} + 30t \hat{j}\)
   • At \(t = 1\), \(\vec{a} = 20 \hat{i} + 30 \times 1 \hat{j} = 20 \hat{i} + 30 \hat{j}\)
   • \(\vec{F} = m \cdot \vec{a} = 0.1 \times (20 \hat{i} + 30 \hat{j}) = 2 \hat{i} + 3 \hat{j}\)
4. Calculate Angular Momentum:
   • Angular momentum \(\vec{L} = \vec{r} \times \vec{p}\)
   • At \(t = 1\), \(\vec{r} = 10 \times 1^2 \hat{i} + 5 \times 1^3 \hat{j} = 10 \hat{i} + 5 \hat{j}\)
   • \(\vec{L} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 10 & 5 & 0 \\ 2 & 1.5 & 0 \end{vmatrix} = (0 - 0)\hat{i} - (0 - 0)\hat{j} + (10 \times 1.5 - 5 \times 2)\hat{k} = 15 - 10\hat{k} = 5\hat{k}\)
5. Calculate Torque:
   • Torque \(\vec{\tau} = \vec{r} \times \vec{F}\)
   • \(\vec{\tau} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 10 & 5 & 0 \\ 2 & 3 & 0 \end{vmatrix} = (0 - 0)\hat{i} - (0 - 0)\hat{j} + (10 \times 3 - 5 \times 2)\hat{k} = 30 - 10\hat{k} = 20\hat{k}\)
6. Conclusion: All four options are correct. Hence, the answer is: A, B, C, D are correct.