Class 11 > Unit # 04: Rotational & Circular Motion > Rotational Quantities

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Unit No. 4 Rotational and Circular Motion - Angular Displacement, Velocity, and Acceleration

Define Angular Displacement, Angular Velocity and Angular Acceleration. Derive relation between Rotational Quantities.

Angular Displacement

"The angle through which a body moves while traveling along a circular path is called angular displacement. It is the angle subtended at the center during angular motion."

Angular displacement is measured in degrees and radians.

• (i) Degree: When a rotating object completes one full revolution, it subtends an angle of $360^\circ$ at the center of its circular path, making its angular displacement $360^\circ$.
• (ii) Radian: It is the angle subtended at the center of a circle by an arc whose length is equal to the radius of the circle.

Relation Between Radian and Degree

Consider a circle of radius $r$ with center $O$. An arc $AB$ of length equal to $r$ subtends $1\text{ radian}$ at the center. Since the total circumference of the circle is equal to $2\pi r$:

• An arc of length $r$ subtends $1\text{ radian}$.
• The entire circumference ($2\pi r$) subtends $2\pi\text{ radians}$ at the center.

Since the entire circumference also subtends $360^\circ$ at the center:

$2\pi \text{ radians} = 360^\circ \implies \pi \text{ radians} = 180^\circ$

$1 \text{ radian} = \frac{180^\circ}{\pi} \approx 57.3^\circ$

Relation Between Linear and Angular Displacement

The arc length ($S$) is directly proportional to the angle subtended ($\theta$) at the center:

$S = r\theta \quad \text{or} \quad \theta = \frac{S}{r}$

Where $r$ is the radius of the circle and $\theta$ is measured in radians.

Angular Velocity

"The rate of change of angular displacement is called angular velocity, denoted by $\omega$ (omega)."

If $\Delta\theta$ is the change in angular displacement during a time interval $\Delta t$, the average angular velocity is:

$\omega_{\text{avg}} = \frac{\Delta\theta}{\Delta t}$

As the time interval approaches zero ($\Delta t \to 0$), the instantaneous angular velocity is given by:

$\omega = \lim_{\Delta t \to 0} \frac{\Delta\theta}{\Delta t} = \frac{d\theta}{dt}$

Unit of Angular Velocity

Angular velocity is measured in $\text{deg/sec}$, $\text{rad/sec}$, or $\text{rev/sec}$. The SI unit is $\text{rad/sec}$ (or $\text{sec}^{-1}$ since radians are dimensionless).

• $1 \text{ revolution} = 2\pi \text{ rad}$
• $1 \text{ rpm} = \frac{2\pi}{60} \text{ rad/sec} \approx 0.105 \text{ rad/sec}$

Direction of Angular Velocity

The direction of angular velocity is always along the axis of rotation. According to the Right-Hand Rule, if the fingers of the right hand curl in the direction of rotation, the extended thumb points in the direction of $\omega$ (out of the page for counter-clockwise rotation, and into the page for clockwise rotation).

Angular Acceleration

"The rate of change of angular velocity is called angular acceleration, denoted by $\alpha$ (alpha)."

If $\Delta\omega$ is the change in angular velocity during a time interval $\Delta t$, the average angular acceleration is:

$\alpha_{\text{avg}} = \frac{\Delta\omega}{\Delta t}$

The instantaneous angular acceleration for an infinitesimally small time interval is:

$\alpha = \lim_{\Delta t \to 0} \frac{\Delta\omega}{\Delta t} = \frac{d\omega}{dt}$

Unit of Angular Acceleration

The SI unit of angular acceleration is $\text{rad/sec}^2$.

Direction of Angular Acceleration

The direction of $\alpha$ is parallel to $\omega$ if angular velocity is increasing, and antiparallel (opposite) to $\omega$ if angular velocity is decreasing along the axis of rotation.

Relation Between Rotational Quantities

Consider an object performing both linear and rotational motion along a circular path of radius $r$:

1. Relation Between Linear and Angular Velocity ($v = r\omega$):

• Linear displacement relation: $S = r\theta \quad \text{--- (i)}$
• Angular velocity definition: $\omega = \frac{\Delta\theta}{\Delta t} \quad \text{--- (ii)}$
• Linear velocity definition: $v = \frac{\Delta S}{\Delta t} \quad \text{--- (iii)}$

Substituting equation (i) into equation (iii):

$v = \frac{\Delta(r\theta)}{\Delta t} = r \frac{\Delta\theta}{\Delta t}$

Since $\frac{\Delta\theta}{\Delta t} = \omega$:

$v = r\omega$

2. Relation Between Linear and Angular Acceleration ($a = r\alpha$):

Starting with the velocity relation $v = r\omega$ and dividing by the time interval $\Delta t$ on both sides:

$\frac{\Delta v}{\Delta t} = r \frac{\Delta\omega}{\Delta t}$

Since linear acceleration $a = \frac{\Delta v}{\Delta t}$ and angular acceleration $\alpha = \frac{\Delta\omega}{\Delta t}$:

$a = r\alpha$

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