Q115 / 150 · MHT CET 2025 (23 Apr Shift 2) · Physics

What is the linear velocity if angular velocity $\vec{\omega}=3 \hat{i}-4 \hat{j}+\hat{k}$ and radius $\vec{r}=(5 \hat{i}-6 \hat{j}+6 \hat{k})$ ?

  • A
    $(-30 \hat{\mathrm{i}}-13 \hat{\mathrm{j}}-38 \hat{\mathrm{k}})$
  • B
    $(8 \hat{\mathrm{i}}-10 \hat{\mathrm{j}}+7 \hat{\mathrm{k}})$
  • Correct answer
    C
    $(-18 \hat{\mathrm{i}}-13 \hat{\mathrm{j}}+2 \hat{\mathrm{k}})$
  • D
    $(-2 \hat{i}-2 \hat{j}-5 \hat{k})$

Solution

The linear velocity is obtained from the cross product $\vec{v} = \vec{\omega} \times \vec{r}$ where $\vec{\omega} = 3\hat{i} - 4\hat{j} + \hat{k}$ and $\vec{r} = 5\hat{i} - 6\hat{j} + 6\hat{k}$.

Computing the determinant:

$\vec{v} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 3 & -4 & 1 \\ 5 & -6 & 6 \end{vmatrix} = \hat{i}(-24 + 6) - \hat{j}(18 - 5) + \hat{k}(-18 + 20) = -18\hat{i} - 13\hat{j} + 2\hat{k}$

This result corresponds to option C.

$\boxed{\text{C}}$

Hence, the correct answer is option: 'C'.

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