Question

A solid body rotates about a stationary axis so that its angular velocity depends on the rotational angle ϕ as \[\omega = {\omega _0} - k\varphi \] where $ω_0$​ and k are positive constants. At the moment t=0,ϕ=0, the time dependence of rotation angle is?
A. \[K{\omega _0}{e^{ - kt}}\]
B. \[\dfrac{{{\omega _0}}}{k}{e^{ - kt}}\]
C. \[\dfrac{{{\omega _0}}}{k}(1 - {e^{ - kt}})\]
D. \[\dfrac{k}{{{\omega _0}}}({e^{ - kt}} - 1)\]

Answer 1

Hint: If a solid body is rotating along a fixed axis. It must be rotating in a circular path with some angular velocity and angular displacement. In question, it is given that about a stationary axis angular velocity depends on the rotational angle ϕ. Time dependence of rotation angle can be determined by making rotational angle ϕ dependent on t and after differentiating it we also get angular velocity \[\omega \] which is also time-dependent. After integrating the equation we easily get the required relationship of the time dependence of the rotation angle.

Formula used
Angular velocity is the rate of change of position angle with respect to time:
\[\omega = \dfrac{{d\varphi }}{{dt}}\]
Where, \[\omega\] is angular velocity, \[d\varphi \] is change in rotational angle and dt is change in time.
\[\int {\dfrac{1}{x}dx = } \] log x + C
Where ln is natural logarithm

Complete step by step solution:
Given angular velocity depends on the rotational angle ϕ as:
\[\omega = {\omega _0} - k\varphi \]
As we know \[\omega = \dfrac{{d\varphi }}{{dt}}\]
Now using this in above equation
\[\dfrac{{d\varphi }}{{dt}} = {\omega _0} - k\varphi \]
\[\Rightarrow \dfrac{{d\varphi }}{{{\omega _0} - k\varphi }} = dt\]

By integrating on both sides, initially time and rotational angle is given as t=0 and \[\varphi \]=0
\[\int\limits_0^\varphi {\dfrac{{d\varphi }}{{{\omega _0} - k\varphi }}} = \int\limits_0^t {dt} \]
\[\Rightarrow - \dfrac{1}{k}\left[ {ln({\omega _0} - k\varphi )} \right]_0^\varphi = \left[ t \right]_0^t\]
\[\Rightarrow - \dfrac{1}{k}\left[ {ln(\dfrac{{{\omega _0} - k\varphi }}{{{\omega _0}}})} \right] = t\]
\[\Rightarrow ln\left( {\dfrac{{{\omega _0} - k\varphi }}{{{\omega _0}}}} \right) = - kt\]
\[\Rightarrow \dfrac{{{\omega _0} - k\varphi }}{{{\omega _0}}} = {e^{ - kt}}\]
\[\Rightarrow {\omega _0} - k\varphi = {\omega _0}{e^{ - kt}}\]
\[\Rightarrow k\varphi = {\omega _0} - {\omega _0}{e^{ - kt}}\]
\[\Rightarrow \varphi = \dfrac{1}{k}\left( {{\omega _0} - {\omega _0}{e^{ - kt}}} \right)\]
\[\therefore \dfrac{{{\omega _0}}}{k}(1 - {e^{ - kt}})\]
Therefore , the time dependence of rotation angle is \[\dfrac{{{\omega _0}}}{k}(1 - {e^{ - kt}})\].

Hence option C is the correct answer

Note: Angular velocity is the concept that is applied to bodies that are moving along a circular path. So we need to understand the concept of rotation. The velocity associated with the rigid body which rotates about a fixed axis is known as angular velocity.