Question

The position of an object having mass \(0.1\) kg as a function of time \(t\) is given as} \[ \vec r = (10t^2\hat{i}+5t^3\hat{j})\ \text{m}. \] At \(t=1\) s, which of the following statements are correct?} A.} Linear momentum \( \vec p = (2\hat{i}+1.5\hat{j})\) kg m/s. B.} Force acting on the object \( \vec F = (2\hat{i}+3\hat{j})\) N. C.} Angular momentum about origin \( \vec L = 15\hat{k}\) J s. D.} Torque about origin \( \vec \tau = 20\hat{k}\) N m. Choose the correct answer.

• A. A, B and C only
• B. B, C and D only
• C. A, C and D only
• D. A, B and D only

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
We differentiate position to find velocity and acceleration. Then we use Newton's second law for force and momentum, and cross products for torque and angular momentum.
Step 2: Key Formula or Approach:
1. \(\vec{v} = d\vec{r}/dt\), \(\vec{a} = d\vec{v}/dt\).
2. \(\vec{p} = m\vec{v}\), \(\vec{F} = m\vec{a}\).
3. \(\vec{L} = \vec{r} \times \vec{p}\), \(\vec{\tau} = \vec{r} \times \vec{F}\).
Step 3: Detailed Explanation:
Calculate \(\vec{v}\) and \(\vec{a}\):
\(\vec{v} = 20t \hat{i} + 15t^2 \hat{j}\). At \(t=1\), \(\vec{v} = 20 \hat{i} + 15 \hat{j}\).
\(\vec{a} = 20 \hat{i} + 30t \hat{j}\). At \(t=1\), \(\vec{a} = 20 \hat{i} + 30 \hat{j}\).
Test statements at \(t=1, \vec{r} = 10 \hat{i} + 5 \hat{j}\):
(A) \(\vec{p} = 0.1(20 \hat{i} + 15 \hat{j}) = 2\hat{i} + 1.5\hat{j}\). (Correct)
(B) \(\vec{F} = 0.1(20 \hat{i} + 30 \hat{j}) = 2\hat{i} + 3\hat{j}\). (Correct)
(C) \(\vec{L} = (10 \hat{i} + 5 \hat{j}) \times (2\hat{i} + 1.5\hat{j}) = (10 \times 1.5 - 5 \times 2)\hat{k} = 5\hat{k} \text{ Js}\). (Incorrect, says 15)
(D) \(\vec{\tau} = (10 \hat{i} + 5 \hat{j}) \times (2\hat{i} + 3\hat{j}) = (10 \times 3 - 5 \times 2)\hat{k} = 20\hat{k} \text{ Nm}\). (Correct)
Step 4: Final Answer:
Statements A, B and D are correct.