Question

What is the value of linear velocity if $\omega = 3\hat i - 4\hat j + \hat k$ and $r = 5\hat i - 6\hat j + 6\hat k$?
A) $6\hat i + 2\hat j - 3\hat k$
B) $ - 18\hat i - 13\hat j + 2\hat k$
C) $4\hat i - 13\hat j + 6\hat k$
D) $6\hat i - 2\hat j + 8\hat k$

Answer 1

Hint: In this question, we need to determine the value of the linear velocity such that the angular velocity and the radius of the body (circular) have been given. For this, we will use the relation between the linear velocity, angular velocity and the radius.

Complete step by step answer:In scalar form, the product of the angular velocity and the radius of the wheel results in the linear velocity of the wheel. Mathematically, $v = \omega r$ where, ‘v’ is the linear velocity of the disc, ‘$\omega $’ is the angular velocity of the disc and ‘r’ is the radius of the disc.
In vector form, the cross product of the angular velocity and the radius of the disc results in the linear velocity of the disc. Mathematically, $\overrightarrow v = \overrightarrow \omega \times \overrightarrow r $ where, $\overrightarrow v $ is the linear velocity of the disc, $\overrightarrow \omega $ is the angular velocity of the disc and $\overrightarrow r $ is the radius of the disc.
Here, in the question, the vector form of the angular velocity and the radius of the disc has been given.
So, substitute $\omega = 3\hat i - 4\hat j + \hat k$ and $r = 5\hat i - 6\hat j + 6\hat k$ in the equation $\overrightarrow v = \overrightarrow \omega \times \overrightarrow r $ to determine the vector form of the linear velocity of the disc.
$
  \overrightarrow v = \overrightarrow \omega \times \overrightarrow r \\
   = \left| {3\hat i - 4\hat j + \hat k} \right| \times \left| {5\hat i - 6\hat j + 6\hat k} \right| \\
   = \left| {\begin{array}
  {\hat i}&{\hat j}&{\hat k} \\
  3&{ - 4}&1 \\
  5&{ - 6}&6
\end{array}} \right| \\
   = \hat i\left( { - 24 + 6} \right) - \hat j\left( {18 - 5} \right) + \hat k\left( { - 18 + 20} \right) \\
   = - 18\hat i - 13\hat j + 2\hat k \\
$
Hence, the linear velocity of the disc for which the angular velocity is $\omega = 3\hat i - 4\hat j + \hat k$ and the radius is $r = 5\hat i - 6\hat j + 6\hat k$ is $ - 18\hat i - 13\hat j + 2\hat k$.
Option B is correct.

Note:Here, in the question, we have used the vector form as both the angular velocity and the radius of the disc have been given in the vector form only. Moreover, the answer, i.e., the linear velocity is also in vector form.