Comprehensive Study Notes on Angular Motion

Introduction to Angular Motion

• Definition: Angular motion is the movement of an object around a fixed point or axis. It is also commonly referred to as rotational motion.

• Complex Forms/Examples:

  • Wheel turning.

  • Gear rotation.

  • Flywheel rotating.

  • Motor shaft rotation.

Key Definitions and Formulas

• Angular Displacement ($\theta$):

  • Definition: The angle through which an object rotates.

  • Unit: Radians ($\text{rad}$).

  • Formula:     $\theta = \frac{s}{r}$

  • Variables:

    • $\theta$ = angular displacement ($\text{rad}$)

    • $s$ = arc length ($\text{m}$)

    • $r$ = radius ($\text{m}$)

• Angular Velocity ($\omega$):

  • Definition: The rate of change of angular displacement.

  • Unit: Radians per second ($\text{rad/s}$).

  • Formula:     $\omega = \frac{\theta}{t}$

  • Formula using Revolutions per Minute ($N$):     $\omega = \frac{2\pi \times N}{60}$

  • Variable:

    • $N$ = revolution per minute ($\text{r/min}$)

• Angular Acceleration ($\alpha$):

  • Definition: The rate of change of angular velocity.

  • Unit: Radians per second squared ($\text{rad/s}^2$).

  • Formula:     $\alpha = \frac{\omega_f - \omega_i}{t}$

  • Variables:

    • $\omega_f$ = final angular velocity ($\text{rad/s}$)

    • $\omega_i$ = initial angular velocity ($\text{rad/s}$)

    • $t$ = time ($\text{s}$)

Relationship Between Linear and Angular motion

• Linear Velocity ($v$):

  • Formula:     $v = r \times \omega$

  • Variables:

    • $v$ = linear velocity ($\text{m/s}$, also written as \text{m\cdot s}^{-1})

    • $r$ = radius ($\text{m}$)

• Linear Acceleration ($a$):

  • Formula:     $a = r \times \alpha$

Equations of Angular Motion

These equations represent the kinematic relationships in rotational motion, mirroring linear motion equations:

1. $\omega_f = \omega_i + \alpha \times t$

2. $\theta = \omega_i \times t + \frac{1}{2} \times \alpha \times t^2$

3. $\omega_f^2 = \omega_i^2 + 2 \times \alpha \times \theta$

Torque (Turning Effect)

• Definition: Torque is the ability of a force to rotate an object.

• Unit: Newton-meters ($\text{Nm}$).

• Formula:     $T = F \times s$

• General Formula:     $T = F \times s \times \sin(\theta)$

• Variables:

  • $T$ = torque ($\text{Nm}$)

  • $F$ = force ($\text{N}$)

  • $s$ = distance ($\text{m}$)

• Conditions for Increasing Torque:

  • Increase the Force ($F$).

  • Increase the Distance from the pivot ($s$).

Moment of Inertia

• Definition: Resistance of an object to rotational motion.

• Unit: Kilogram meters squared (\text{kg\cdot m}^2).

• General Formula:     $I = m \times r^2$

• Variables:

  • $I$ = moment of inertia (\text{kg\cdot m}^2)

  • $m$ = mass ($\text{kg}$)

  • $r$ = radius of gyration ($\text{m}$)

• Common Shapes:

  • Solid Disc:     $I = \frac{1}{2} \times m \times r^2$

  • Thin Ring:     $I = m \times r^2$

Work Done

• In Linear Motion:

  • Definition: Work done is when a force moves an object through a distance.

  • Formula:     $W = F \times s$

  • Variables:

    • $W$ = work done ($\text{J}$)

    • $F$ = force ($\text{N}$)

    • $s$ = distance moved ($\text{m}$)

• In Angular Motion:

  • Formula:     $W = T \times \theta$

  • Variables:

    • $W$ = work done ($\text{J}$)

    • $T$ = torque ($\text{Nm}$)

    • $\theta$ = angular displacement ($\text{rad}$)

Power

• In Linear Motion:

  • Definition: The rate at which work is done.

  • Formula 1:     $P = \frac{W}{t}$

  • Formula 2:     $P = F \times v$

  • Variables:

    • $P$ = power ($\text{W}$)

    • $W$ = work ($\text{J}$)

    • $t$ = time ($\text{s}$)

    • $F$ = force ($\text{N}$)

    • $v$ = velocity ($\text{m/s}$, also written as \text{m\cdot s}^{-1})

• In Angular Motion:

  • Formula 1:     $P = T \times \omega$

  • Formula 2 (using RPM):     $P = \frac{2\pi \times N \times T}{60}$

  • Variables:

    • $P$ = Power ($\text{W}$)

    • $T$ = Torque ($\text{Nm}$)

    • $\omega$ = angular velocity ($\text{rad/s}$)

    • $N$ = revolutions per minute ($\text{r/min}$)

Unit Conversions: Degrees and Radians

• Converting Degrees to Radians:     $\text{radians} = \text{degrees} \times \frac{\pi}{180}$

  • Example: Convert $90^\circ$ to radians:     $90 \times \frac{\pi}{180} = \frac{\pi}{2}\,\text{rad}$

• Converting Radians to Degrees:     $\text{degrees} = \text{radians} \times \frac{180}{\pi}$

  • Example: Convert $2\,\text{rad}$ to degrees:     $2 \times \frac{180}{\pi} = 114.592^\circ$