A force of $(2 \hat{i}+3 \hat{j}+4 \hat{k}) N$ acts on a particle whose position vector with respect to the…

A force of $(2 \hat{i}+3 \hat{j}+4 \hat{k}) N$ acts on a particle whose position vector with respect to the origin of the coordinate system is $(6 \hat{i}+b \hat{j}+12 \hat{k}) \mathrm{m}$. If the angular momentum of the body is constant, the value of ' $b$ ' is
  1. 6
  2. 9
  3. 12
  4. 3

Solution

Torque, $\vec{\tau}=\frac{\mathrm{d} \overrightarrow{\mathrm{L}}}{\mathrm{dt}}$ and $\dot{\vec{\tau}}=0$ when $\overrightarrow{\mathrm{L}}=$ constant $ \begin{aligned} & \therefore \vec{\tau}=\overrightarrow{\mathrm{r}} \times \overrightarrow{\mathrm{F}}=0 \text { or, }(6 \hat{\mathrm{i}}+\mathrm{b} \hat{\mathrm{j}}+12 \hat{\mathrm{k}}) \times(2 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}+4 \hat{\mathrm{k}})=0 \\ & \quad \text { or, } 4 \mathrm{~b}-36=0 \quad \therefore \mathrm{b}=\frac{36}{4}=9 \end{aligned} $

Asked in: AP EAMCET 2023 (15 May Shift 1)

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