JEE Mains · Physics · STD 11 - System of particles and rotational motion
- JEE Mains 2025
- Hard
- Numerical
The position vectors of two 1 kg particles, (A) and (B), are given by
\(\overrightarrow{\mathrm{r}}_{\mathrm{A}}=\left(\alpha_1 \mathrm{t}^2 \hat{i}+\alpha_2 \mathrm{t} \hat{j}+\alpha_3 \mathrm{t} \hat{k}\right) \mathrm{m}\) and \(\overrightarrow{\mathrm{r}}_{\mathrm{B}}=(\beta_1 \mathrm{t} \hat{i}+\beta_2 \mathrm{t}^2 \hat{j}\) \(+\beta_3 \mathrm{t} \hat{k}) \mathrm{m}\), respectively; \((\alpha_1=1 \mathrm{~m} / \mathrm{s}^2, \alpha_2=3 \mathrm{n} \mathrm{m} / \mathrm{s},\) \(\alpha_3=2 \mathrm{~m} / \mathrm{s},\) \(\beta_1=2 \mathrm{~m} / \mathrm{s}, \beta_2=-1 \mathrm{~m} / \mathrm{s}^2, \beta_3=4 \mathrm{pm} / \mathrm{s})\), where t is time, n and p are constants. At \(t=1 \mathrm{~s},\left|\overrightarrow{V_A}\right|=\left|\vec{V}_B\right|\) and velocities \(\vec{V}_A\) and \(\vec{V}_B\) of the particles are orthogonal to each other. At \(t=1 \mathrm{~s}\), the magnitude of angular momentum of particle (A) with respect to the position of particle (B) is \(\sqrt{\mathrm{L}} \mathrm{kgm}^2 \mathrm{~s}^{-1}\). The value of L is _______ .
Pick an option to check your answer.
Answer & Solution
Correct Answer
(D) 90
Step-by-step Solution
Detailed explanation
At \(t=1\)
\(r_{A B}=-1 \hat{i}+(3 n+1) \hat{j}+(2-4 p) \hat{k}\)
At \(t=1\)
\(v_A=2 \hat{i}+3 n \hat{j}+2 \hat{k} \)
\(v_B=2 \hat{i}-2 \hat{j}+4 p \hat{k} \)
\(\vec{v}_A-\vec{v}_B=0, \quad 4-6 n+8 p=0\)
\(\left|v_A\right|=\left|v_B\right| \quad(3 n)^2+4=4+16 p^2 \)
\( 3 n=-4 p \)
\( 4+16 p=0 \)
\( p=-\frac{1}{4}, n=\frac{1}{3} \)
\( r_{A B}=-\hat{i}+2 \hat{j}+3 \hat{k} \)
\( v_A=2 \hat{i}+\hat{j}+2 \hat{k} \)
\( \therefore L=m\left|\vec{r}_{A B} \times \vec{v}_A\right|=90\)
\(r_{A B}=-1 \hat{i}+(3 n+1) \hat{j}+(2-4 p) \hat{k}\)
At \(t=1\)
\(v_A=2 \hat{i}+3 n \hat{j}+2 \hat{k} \)
\(v_B=2 \hat{i}-2 \hat{j}+4 p \hat{k} \)
\(\vec{v}_A-\vec{v}_B=0, \quad 4-6 n+8 p=0\)
\(\left|v_A\right|=\left|v_B\right| \quad(3 n)^2+4=4+16 p^2 \)
\( 3 n=-4 p \)
\( 4+16 p=0 \)
\( p=-\frac{1}{4}, n=\frac{1}{3} \)
\( r_{A B}=-\hat{i}+2 \hat{j}+3 \hat{k} \)
\( v_A=2 \hat{i}+\hat{j}+2 \hat{k} \)
\( \therefore L=m\left|\vec{r}_{A B} \times \vec{v}_A\right|=90\)
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