Question

How to find frequency of rotational motion without knowing radius ?
\[{v_1} = 3\,m/s,\,\,{v_2} = 2\,m/s,\,\,{r_2} = {r_1} - 10\,cm\]

Answer 1

Hint: A body is said to be in rotational motion if it rotates on an axis. Then the radius vectors from the axis to all particles undergo the same angular displacement at the same time. If the rigid body in such a motion rotates about a fixed axis that is perpendicular to a fixed plane then it is considered as pure rotational motion.

Formula used:
\[\omega = \dfrac{v}{r}\]
Where\[\omega \] is a constant angular frequency around some axis of rotation,\[v\] is the linear speed at that point and\[r\] is the distance from the axis of rotation.

Complete step by step answer:
We have given that linear speeds are \[{v_1}{\text{ }} = {\text{ }}3\,m/s\] and \[{v_2}{\text{ }} = {\text{ }}2\,m/s\] at the two points 1 and 2 on the rotating body. Also we have given that the radii \[{r_1}\] and \[{r_2}\] are related as \[{r_2}{\text{ }} = {\text{ }}{r_1} - 10{\text{ }}cm\].Now for the rigid rotation we know the formula relating to Angular frequency, radius and the linear speed is given by:
\[\omega = \dfrac{v}{r}\]
And so for the same body we have
\[\omega = \dfrac{{{v_1}}}{{{r_1}}} = \dfrac{{{v_2}}}{{{r_2}}}\]
Using the relation \[{r_2}{\text{ }} = {\text{ }}{r_1} - 10{\text{ }}cm\]
We get:
\[\dfrac{{{v_1}}}{{{r_1}}} = \dfrac{{{v_2}}}{{{r_1} - 10{\text{ }}cm}}\]
On cross multiplying and simplifying for\[{r_1}\] we get,
\[{v_1}\left( {{r_1} - 10} \right) = {v_2}{r_1} \\
\Rightarrow \left( 3 \right)\left( {{r_1} - 10} \right) = \left( 2 \right){r_1} \\
\Rightarrow 3{r_1} - 30 = 2{r_1} \\
\Rightarrow {r_1} = 30\,cm \]
Converting the unit of\[{r_1}\] in S.I. unit meter
\[{r_1} = \dfrac{{30}}{{100}}m\,\, = \,\,0.3\,m\]
Now putting the value of \[{r_1}\] in above relation \[\omega = \dfrac{{{v_1}}}{{{r_1}}}\]we get,
\[\omega = \dfrac{3}{{0.3}}\,rad/s \\
\therefore \omega = 10\,rad/s \\ \]
Hence the angular frequency we obtained is \[\omega = 10\,rad/s\].

Note: The usual units should be kept in mind for mistakes related to units and conversion. One should get to know the difference between angular velocity and linear velocity, here in this question we are provided with linear speed. The S.I unit of general frequency for waves is Hertz but. Unit for frequency in rotational motion is Radian per second.