Question

Which of the following relations is not correct?
(A) \[\overrightarrow V = \overrightarrow \omega \times \overrightarrow r \]
(B) \[\overrightarrow V = \overrightarrow r \times \overrightarrow \omega \]
(C) \[\overrightarrow {\delta s} = \overrightarrow {\delta \theta } \times \overrightarrow r \]
(D) \[V = r\omega \]

Answer 1

Hint: This given problem can be solved using the concept of linear velocity on the account of rotation i.e., circular motion of any object along the path that is the circumference of a circle.

Complete step by step answer:
Step 1: Linear velocity: Linear velocity can be defined as the rate of change of position of an object that is moving along a straight path.
It can be defined by the formula given –
\[\overrightarrow V = \dfrac{{\overrightarrow {dS} }}{{dt}}\] …………..(1)
 where \[\overrightarrow {dS} = \]change of linear displacement, \[dt = \]time take for \[\overrightarrow {dS} \]
It is denoted by V and is measured in meters per second i.e., m/s.
Angular velocity: On the other hand, angular velocity can be defined as the rate of change of the angular displacement over the time taken by that object for that much displacement. Angular velocity applies to move objects along a circular path only.
It can be defined by the formula given –
\[\overrightarrow \omega = \dfrac{{\overrightarrow {d\theta } }}{{dt}}\] …………..(2)
 where \[\overrightarrow {d\theta } = \]change of angular displacement, \[dt = \]time taken for \[\overrightarrow {d\theta } \]
It is denoted by \[\omega \] and is measured in radians per second i.e., rad/s.
Angular velocity is a vector quantity and direction is the same as that of \[d\theta \]

Step 2:
The relationship between linear velocity and angular velocity is given by-
\[\overrightarrow V = \overrightarrow \omega \times \overrightarrow r \] …………..(3)
If values of \[\overrightarrow V \] and \[\overrightarrow \omega \] is kept in equation (3) from equation (1) and (2), we will get-
\[\overrightarrow {dS} = \overrightarrow {d\theta } \times \overrightarrow r \] …………..(4)

And the magnitude of equation (4) that will be a scalar quantity can be written as –
\[V = r\omega \] …………..(5)

Step 3: Let us consider two vectors \[\overrightarrow A \] and \[\overrightarrow B \], and if we are taking the vector product of these two quantities then we know that vector product does not follow the commutative property i.e.,
\[\overrightarrow A \times \overrightarrow B \ne \overrightarrow B \times \overrightarrow A \]
But \[\overrightarrow A \times \overrightarrow B = - \overrightarrow B \times \overrightarrow A \]
So, from equation (3) and the above property, we can say that –
\[\overrightarrow V = - \overrightarrow r \times \overrightarrow \omega \] …………..(6)
So, from equations (3), (4), (5), and (6), we can say that the information in option (B) is wrong.
So, the required answer is in option (B).

$\therefore $ The correct option is (B).

Note:
It is important to remember that nothing actually moves in the direction of the angular velocity vector. The direction of \[\overrightarrow \omega \] simply means that the rotational motion of the object taking place in a plane that is perpendicular to \[\overrightarrow \omega \].
The direction of angular velocity can be determined by the right-hand rule.