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Physics · SYSTEMS OF PARTICLES AND ROTATIONAL MOTION
हिन्दीA rigid body rotates about a fixed axis with angular velocity \( \vec{\omega} = \omega\hat{k} \). If the position vector of a point \( P \) on the body from a point \( O \) on the rotation axis is \( \vec{r} = 3\hat{i} + 4\hat{j} + 5\hat{k} \), the ratio of the linear speed of point \( P \) to the product \( \omega |\vec{r}| \) is:
Correct answer · Option 1
\( \frac{1}{\sqrt{2}} \)
Explanation
Distance from rotation axis (z-axis) is \( r_\perp = \sqrt{3^2 + 4^2} = 5 \). Magnitude of position vector is \( |\vec{r}| = \sqrt{3^2 + 4^2 + 5^2} = \sqrt{50} = 5\sqrt{2} \). Linear speed \( v = \omega r_\perp = 5\omega \). Ratio is \( \frac{v}{\omega |\vec{r}|} = \frac{5\omega}{5\sqrt{2}\omega} = \frac{1}{\sqrt{2}} \).
