# Circular Motion IIT JEE Study Material

The circular motion is a motion in which the distance of a particle (radius) moving in a plane from a fixed point (center) remains constant then its motion. The Circular motion is a 2D motion in a plane. It can be classified into two types: uniform with constant speed and constant angular rotation rate or non-uniform with changing rotation rate.

## Variables of Motion:

Angular Position: The angle subtended by the position vector with the reference line is called the angular position of a particle.

Angular Displacement $\mathbf{\left ( \theta \right )}$: It is the angle by which the position vector of the moving particle is rotated with respect to the reference line. It is the dimensionless quantity and its SI unit is radian. The angular displacement is also measured in degrees or revolutions.

$\mathbf{2\;\pi \;rad\;=\;360^{0}\;=\;1\;rev}$

Angular Velocity $\mathbf{\left ( \omega \right )}$: If $\mathbf{ \theta _{1}}$ and $\mathbf{ \theta _{2}}$ are angular positions of a particle at time $\mathbf{t_{1}}$
and $\mathbf{t_{2}}$
respectively, then, the average angular velocity is given by:

$\mathbf{\omega _{av}\;=\;\frac{Total\;angle\;of\;rotation}{Total\;time\; taken}\;=\;\frac{ \theta _{2}\;-\; \theta _{1}}{t_{2}\;-\;t_{1}}\;=\;\frac{\Delta \theta }{\Delta t}}$

A. Instantaneous Angular Velocity i.e. $\mathbf{\omega \;=\;\lim_{\Delta t \rightarrow 0}\;\frac{\Delta \; \theta }{\Delta \;t}\;=\;\frac{d \theta }{dt}}$

B. If $\mathbf{ \theta \;=\;a\;-\;bt\;+\;ct^{2}\;\;then\;\;\omega \;=\;\frac{d \theta }{dt}\;=\;-\;b\;+\;2\;ct}$

C. If a body makes ‘n’ rotations in time t seconds then, the angular velocity:

$\mathbf{\omega _{av}\;=\;\frac{2\;\pi \;n}{t}}$ radian $\mathbf{sec^{-1}}$.

D. It is an axial vector with dimensions $\mathbf{T^{-1}}$and S.I. unit = rad $\mathbf{sec^{-1}}$.

Angular Acceleration $\mathbf{\left ( alpha \right )}$: If $\mathbf{\omega _{1}}$ and $\mathbf{\omega _{2}}$ are angular velocities of a particle at time $\mathbf{t_{1}}$
and $\mathbf{t_{2}}$ respectively, then, the average angular velocity is given by:

$\mathbf{\alpha _{av}\;=\;\frac{\omega _{2}\;-\;\omega _{1}}{t_{2}\;-\;t_{1}}\;=\;\frac{\Delta \omega}{\Delta t}}$

A. Instantaneous Angular Acceleration i.e. $\mathbf{\alpha \;=\;\lim_{\Delta t \rightarrow 0}\;\frac{\Delta \;\omega }{\Delta \;t}\;=\;\frac{d\;\omega }{dt}\;=\;\omega \;\;\frac{\Delta \;\omega}{\Delta \; \theta }}$

B. The motion is said to be the uniform circular motion if, $\mathbf{alpha\;=\;0}$

C. $\mathbf{\omega \;=\;\frac{d \theta }{dt}\;\;and\;\;\alpha \;=\;\frac{d\;\omega }{dt}\;=\;\frac{d^{2}\; \theta }{dt^{2}}}$

D. It is an axial vector with dimensions as $\mathbf{T^{-2}}$
and unit as rad $\mathbf{sec^{-2}}$

## Uniform circular motion and Non-Uniform Circular Motion:

If the particle moves in the circle with a uniform speed we call it a uniform circular motion. In this case, the tangential component of acceleration is absent.

$\mathbf{\frac{dv}{dt}}$ = 0 or $\mathbf{a_r}$ = $\mathbf{\omega^2 r}$

In a non-uniform circular motion, the speed of the particle moving in a circle is not constant. In this case, the tangential component of acceleration is present.

$\mathbf{\frac{dv}{dt}}$ = $\mathbf{a_t}$ or $\mathbf{a_r}$ = $\mathbf{\omega^2 r}$

In case of non-uniform circular motion, the total acceleration is the resultant of the tangential and the radial acceleration.

## The equations of motion associated with the circular motion:

A. $\mathbf{ \theta}$ = $\mathbf{\omega_0 t+\frac{1}{2}at^2}$

B. $\mathbf{\omega}$ = $\mathbf{\omega_0+at}$

C. $\mathbf{\omega^2}$ = $\mathbf{\omega_0^2+2\alpha \theta}$

where, $\mathbf{ω_0}$ and $\mathbf{ω}$ are the angular velocities at t = 0 and at time t respectively, and $\mathbf{ \theta}$ is the angular position at any time.

## Velocity and Acceleration of a Particle in Circular Motion:

Let, at = tangential acceleration and $\mathbf{a_{c}}$ = Centripetal Acceleration, therefore, the net acceleration of the particle will be:

$\mathbf{a=\sqrt{a_{c}^{2}\;+\;a_{t}^{2}}=\sqrt{\left ( r\;\omega ^{2} \right )^{2}\;+\;\left ( \frac{dv}{dt} \right )^{2}}=\sqrt{\left ( \frac{v^{2}}{r} \right )^{2}\;+\;\left ( \frac{dv}{dt} \right )^{2}}}$

Where, v is the linear speed of the particle = r $\mathbf{\omega }$ = Relationship between speed and angular velocity.

And, $\mathbf{\frac{dv}{dt}}$ is the rate of change of speed.

## Relative Angular Velocity:

The Relative angular velocity of an object P w.r.t the other moving object Q is the angular velocity of the position vector of P w.r.t. Q.

$\mathbf{\omega_{ab}\;=\;\frac{Relative\;\;velocity\;\;of\;\;P\;\;w.r.t.\;\;Q \;\;perp \;\;to \;\;line\;\;PQ}{Distance\;\;between\;\;P\;\;and\;\;Q}}$

i.e. $\mathbf{\omega _{PQ}\;=\;\frac{v_{P}\;Sin\; \theta _{1}\;+\;v_{Q}\;Sin\; \theta _{2}}{r}}$

## Dynamics of Circular Motion:

In a circular motion, Newton’s law is applied in two perpendicular directions. The first one is along the tangent and the other one is perpendicular to it (towards the center). The component of force acting along the tangent is called the tangential force (Ft) and the component of force acting along the center is called the centripetal force (Fc).

Tangential Force $\mathbf{F_{t}\;=\;M\;a_{t}\;=\;M\;\frac{dv}{dt}\;=\;M\;\alpha \;r}$

Centripetal Force $\mathbf{F_{c}\;=\;M\;a_{t}\;=\;m\;\omega ^{2}\;r\;=\;\frac{m\;v^{2}}{r}}$

Note: In absence of the centripetal force the object will move in a straight line with constant speed.

The radius of curvature of the instantaneous circle (R):

$\mathbf{R\;=\;\frac{\left [ 1\;+\;\left ( \frac{dy}{dx} \right )^{2} \right ]^{\frac{3}{2}}}{\left | \frac{d^{2}y}{dx^{2}} \right |}}$

## Motion in a vertical Circle:

Consider the motion of a stone tied to a string and whirled in a vertical circle. If at any time (t) the body is at the angular position θ. Now, the forces acting are: tension (T) in the string along the radius (r) towards the center and the weight (w) of the body i.e mg acting vertically downwards.

Now, Applying Newton’s law towards centre we get,

$\mathbf{T\;-\;mg\;cos\; \theta \;=\;\frac{m\;v^{2}}{r}}$

Or, $\mathbf{T\;=\;\;mg\;cos\; \theta \;+\;\frac{m\;v^{2}}{r}}$

Now, if $\mathbf{T_{min}}$ > 0, the stone will move on the circular path.

And, if $\mathbf{T_{min}}$ ≤ 0, the string will slack and the body will fall down instead of moving on the circle.

Therefore, for looping the loop:

$\mathbf{\frac{m\;v_{H}^{2}}{r}\;-\;mg\;\geq \;0\;\;or\;\;v_{H}\;\geq \;\sqrt{gr}}$

Now applying the conservation of mechanical energy between the lowest point (L) and the highest point (H):

$\mathbf{v_{L}\;\geq \;\sqrt{5\;gr}}$

Therefore, for looping the loop the velocity at the lowest point must be greater than or equal to $\mathbf{\sqrt{5\;gr}}$.