Question

If you have \[10.8\] meters per second, how will you convert that to radians per second?

Answer 1

Hint: Here, we will use the relation between the angular speed and the linear speed for converting the given linear speed value into the corresponding angular speed. For this, we will assume a particular value for the radius and find the required value.

Formula used:
We will use the formula \[\omega = \dfrac{v}{r}\], here \[\omega \] is the angular velocity of a body moving on a circular path of radius equal to \[r\] with a linear or the tangential speed of \[v\].

Complete step-by-step solution:
The unit of the value \[10.8\] given in the above question, which is meters per second, suggests that it is the value of the speed.
We have to convert it to radian per second. We know that the unit radian per second is the unit of the angular speed. This means that we have to use the relation between the angular and the linear speed of an object moving in a circular path.
We know that the angular speed of an object is related to its tangential speed by the relation \[\omega = \dfrac{v}{r}\].
So we will assume that an object is moving on a circular path with the linear or the tangential speed of \[10.8\] meters per second.
Let the radius of the circular path in which the object is moving be equal to \[R\] meters.
Therefore, on substituting \[v = 10.8m{s^{ - 1}}\] and \[r = R\] in the above equation, we get the angular speed as
\[\omega = \dfrac{{10.8}}{R}rad/s\]

Hence, the value of metres per second is equal to \[\dfrac{{10.8}}{R}\] radians per second.

Note:
We know that linear speed is defined as the speed of an object while covering the distance of a linear path. When the object is moving in angular motion then the speed with which the object moves is called angular speed. We need to choose the unit of the radius in such a manner that the unit of the angular speed becomes the second inverse. For example, if the unit of the speed was given to be millimeters per second, then we would have taken the unit of the radius to be millimeters.