Unit 6 : Circular Motion Class 11 Notes For NEB | NEPALI EDUCATE

Circular Motion

The type of motion in which the body is displaced continuously, but is always confined at a fixed distance from a fixed point is called circular motion.

Example: Turning of a vehicle in bending roads, motion of satellites arounds planets, motion of planets around sun etc.

Types of Circular Motion:

1) Uniform Circular Motion:

The circular motion in which an object covers equal angular displacement in equal time interval is called uniform circular motion.

The work done in uniform circular motion is zero due to angle between force and displacement is 90^ο.

2) Non-Uniform Circular Motion:

The circular motion in which an object covers equal angular displacement in unequal time interval is called non-uniform circular motion.

Some terminology Used in Circular Motion:

Angular Displacement (θ):

The angle covered by the body from its initial position to final position in a given time frame is called angular displacement.

In the figure

∠AOB = θ is angular displacement.

Unit of angular displacement is radian represented by rad.

Angular Velocity:

The time rate of change of angular displacement is called angular velocity.

It is denoted by ω.

Its unit is rad s^- (rad per seconds).

Average angular Velocity:

The ration of the angular displacement to the time taken by the particle to undergoes this displacement is called average angular velocity.

It is denoted by ω_av.

Mathematically,

ω_av = $\frac{{{\theta _2} - {\theta _1}}}{{{t_2} - {t_1}}}$

Instantaneous angular velocity:

The limiting value of every angular velocity of the particle in a small time interval such that the time interval approaches to 0 is called instantaneous angular velocity.

It is denoted by ω_ins.

Mathematically,

${\omega _{ins}} = \mathop {\lim }\limits_{\Delta t \to 0} \left( {\frac{{\Delta \theta }}{{\Delta t}}} \right) = \frac{{d\theta }}{{dt}}$

Angular Acceleration:

The rate of change of angular velocity is called angular acceleration.

It is denoted by α.

Its unit is rad s^-2 (radian per second square).

Average angular acceleration:

The ratio of change in angular velocity to the time taken by particle to undergoes the change is called average angular acceleration.

Mathematically,

α_av = ${\alpha _{av}} = \frac{{{\omega _2} - {\omega _1}}}{{{t_2} - {t_1}}} = \frac{{\Delta \omega }}{{\Delta t}}$

Instantaneous angular acceleration:

The limiting value of average angular acceleration as Δt approaches to zero, is called Instantaneous angular acceleration.

It is represented by α_ins.

Mathematically,

${\alpha _{ins}} = \mathop {\lim }\limits_{\Delta t \to 0} \left( {\frac{{\Delta \omega }}{{\Delta t}}} \right) = \frac{{d\omega }}{{dt}}$

Time Period:

Time required by the particle to complete one revolution is called time period. It is denoted by T.

Mathematically, $\omega  = \frac{\theta }{t}$

For One complete revolution, Ɵ=2π and t = T.

$\begin{array}{l} \Rightarrow \omega  = \frac{{2\pi }}{T}\\\therefore T = \frac{{2\pi }}{\omega }\end{array}$

Frequency:

Number of revolutions completed in unit time is called frequency. It is denoted by f.

Mathematically,

$\begin{array}{l}f = \frac{1}{T}\\ \Rightarrow f = \frac{1}{{\frac{{2\pi }}{\omega }}} = \frac{\omega }{{2\pi }}\\\therefore \omega  = 2\pi f\end{array}$

Relationship between Angular Velocity and Linear Velocity:

Let us consider a particle is revolving in circular path of radius r and center O, travers a small angular displacement from A to B in time t as shown in figure. Let v and ω be the linear and angular velocity respectively. 

From Figure,

$\theta $ = $\frac{l}{r}$=$\frac{AB}{r}$ ……..(1)

For Small angle θ, arc length AB = Chord AB. So,

$\overset\frown{AB}$= AB = s (Assume)

Now, Equation (1) becomes

$\theta $ = $\frac{s}{r}$

Or, s = rθ

Differentiating above equation with respect to time,

$\begin{array}{l}\frac{{ds}}{{dt}} = \frac{d}{{dt}}(r\theta )\\i.e.,\;{\rm{v = r}}{\rm{.}}\frac{{d\theta }}{{dt}} = r\omega \\\therefore v = r\omega \end{array}$

Again, differentiating with respect to time, we get

$\begin{array}{l}\frac{{dv}}{{dt}} = \frac{d}{{dt}}(r\omega )\\i.e.,\;{\rm{a = r}}{\rm{.}}\frac{{d\omega }}{{dt}} = r\alpha \\\therefore a = r\alpha \end{array}$ 

Expression for Centripetal Acceleration:

The force required to keep a body in circular path with uniform speed is called centripetal force. This force is always directed towards the center of circular path.

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