A body is rotating with angular velocity ω̅=(3 î-4 ĵ+k̂). The linear velocity of a point having

step 1 In this question, we have to calculate the linear velocity of particle whose angular velocity an

A body is rotating with angular velocity ω̅=(3 î-4 ĵ+k̂)....

A body is rotating with angular velocity $\bar{\omega}=(3 \hat{i}-4 \hat{j}+\hat{k})$. The linear velocity of a point having position vector $\vec{r}=(5 \hat{i}-6 \hat{j}+6 \hat{k})$ is (a) $6 \hat{i}+2 \hat{j}-3 \hat{k}$ (b) $18 \hat{i}+3 \hat{j}-2 \hat{k}$ (c) $-18 \hat{i}-13 \hat{j}+2 \hat{k}$ (d) $6 \hat{i}-2 \hat{j}+8 \hat{k}$

A body is rotating with angular velocity $\bar{\omega}=(3 \hat{i}-4 \hat{j}+\hat{k})$. The linear velocity of a point having position vector $\vec{r}=(5 \hat{i}-6 \hat{j}+6 \hat{k})$ is
(a) $6 \hat{i}+2 \hat{j}-3 \hat{k}$
(b) $18 \hat{i}+3 \hat{j}-2 \hat{k}$
(c) $-18 \hat{i}-13 \hat{j}+2 \hat{k}$
(d) $6 \hat{i}-2 \hat{j}+8 \hat{k}$

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Key Concepts

Angular Velocity

Angular velocity is a vector quantity that represents the rate at which an object rotates about an axis, including both the speed of rotation and the axis direction. It is fundamental to understanding rotational dynamics and helps in describing the rotational state of a system.

Linear Velocity in Rotational Motion

In the context of rotational motion, the linear velocity of a point on a rigid body is determined by its distance from the axis of rotation and the body’s angular velocity. The relationship is given by the formula v = ? × r, where r is the position vector of the point relative to the rotation axis. This formula bridges the gap between rotational and translational motion.

Cross Product

The cross product is a binary operation on two vectors in three-dimensional space. It produces a third vector that is perpendicular to the plane containing the original vectors, with a magnitude equal to the product of the magnitudes of the two vectors and the sine of the angle between them. This operation is key in calculating the linear velocity from a given angular velocity and position vector, as seen in v = ? × r.

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