Question:
A body is rotating with angular velocity $\overrightarrow{\omega}=\left(3\hat{i}-4\hat{j}+\hat{k}\right)$. The linear velocity of a point having position vector $\overrightarrow{r}=\left(5\hat{i}-6\hat{j}+6\hat{k}\right)$ is
A body is rotating with angular velocity $\overrightarrow{\omega}=\left(3\hat{i}-4\hat{j}+\hat{k}\right)$. The linear velocity of a point having position vector $\overrightarrow{r}=\left(5\hat{i}-6\hat{j}+6\hat{k}\right)$ is
Updated On: Jul 5, 2022
- $6\hat{i}+2\hat{j}-3\hat{k}$
- $18\hat{i}+3\hat{j}-2\hat{k}$
- $-18\hat{i}-13\hat{j}+2\hat{k}$
- $6\hat{i}-2\hat{j}+8\hat{k}$
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The Correct Option is C
Solution and Explanation
Here, $\vec{\omega }=3\hat{i}-4\hat{j}+\hat{k}$
$\overrightarrow{r}=5\hat{i}-6\hat{j}+6\hat{k}$
As $\overrightarrow{\upsilon}=\vec{\omega} \times\overrightarrow{r}$
$=\begin{vmatrix}\hat{i}&\hat{j}&\hat{k}\\ 3&-4&1\\ 5&-6&6\end{vmatrix}$
$=\hat{i} (-24-\left(-6\right))+\hat{j} \left(5-18\right)+\hat{k} (-18-\left(-20\right))$
$=-18 \hat{i}-13\hat{j}+2\hat{k}$
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Concepts Used:
System of Particles and Rotational Motion
- The system of particles refers to the extended body which is considered a rigid body most of the time for simple or easy understanding. A rigid body is a body with a perfectly definite and unchangeable shape.
- The distance between the pair of particles in such a body does not replace or alter. Rotational motion can be described as the motion of a rigid body originates in such a manner that all of its particles move in a circle about an axis with a common angular velocity.
- The few common examples of rotational motion are the motion of the blade of a windmill and periodic motion.