🌀 Rotational Kinematics

Angular Quantities

Just as linear motion has position, velocity, and acceleration, rotational motion has corresponding angular quantities.

Angular Position $\theta$

Angle of rotation from reference line:

• Units: radians (rad)
• $2\pi$ rad = 360° = one full rotation
• Conversion: $\theta_{rad} = \theta_{deg} \times \frac{\pi}{180}$

Angular Displacement $\Delta\theta$

Change in angular position: $\Delta\theta = \theta_f - \theta_i$

Units: radians (rad)

Positive: counterclockwise rotation Negative: clockwise rotation

Angular Velocity $\omega$

Rate of change of angular position:

Average: $\omega_{avg} = \frac{\Delta\theta}{\Delta t}$

Instantaneous: $\omega = \frac{d\theta}{dt}$

Units: rad/s (radians per second)

Angular Acceleration $\alpha$

Rate of change of angular velocity:

Average: $\alpha_{avg} = \frac{\Delta\omega}{\Delta t}$

Instantaneous: $\alpha = \frac{d\omega}{dt} = \frac{d^2\theta}{dt^2}$

Units: rad/s² (radians per second squared)

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Analogy with Linear Motion

| Linear Motion | Rotational Motion | |---------------|-------------------| | Position $x$ | Angular position $\theta$ | | Velocity $v$ | Angular velocity $\omega$ | | Acceleration $a$ | Angular acceleration $\alpha$ | | Mass $m$ | Moment of inertia $I$ | | Force $F$ | Torque $\tau$ |

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Relationship Between Linear and Angular

For a point at distance $r$ from axis of rotation:

Arc Length

$s = r\theta$ (where $\theta$ is in radians)

Linear Velocity

$v = r\omega$

Tangential velocity - velocity tangent to circular path

Tangential Acceleration

$a_t = r\alpha$

Component of acceleration tangent to circle (changes speed)

Centripetal Acceleration

$a_c = \frac{v^2}{r} = \omega^2 r$

Component of acceleration toward center (changes direction)

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Rotational Kinematic Equations

For constant angular acceleration $\alpha$:

The Big Four Equations

1. Angular velocity: $\omega_f = \omega_i + \alpha t$

2. Angular displacement: $\theta = \theta_i + \omega_i t + \frac{1}{2}\alpha t^2$

3. Velocity-displacement: $\omega_f^2 = \omega_i^2 + 2\alpha\Delta\theta$

4. Average velocity: $\theta = \theta_i + \frac{1}{2}(\omega_i + \omega_f)t$

💡 These are EXACTLY analogous to linear kinematic equations! Just replace $x \to \theta$, $v \to \omega$, $a \to \alpha$.

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Comparison: Linear vs. Rotational Kinematics

| Linear (constant $a$) | Rotational (constant $\alpha$) | |----------------------|--------------------------------| | $v_f = v_i + at$ | $\omega_f = \omega_i + \alpha t$ | | $x = x_i + v_i t + \frac{1}{2}at^2$ | $\theta = \theta_i + \omega_i t + \frac{1}{2}\alpha t^2$ | | $v_f^2 = v_i^2 + 2a\Delta x$ | $\omega_f^2 = \omega_i^2 + 2\alpha\Delta\theta$ | | $x = x_i + \frac{1}{2}(v_i + v_f)t$ | $\theta = \theta_i + \frac{1}{2}(\omega_i + \omega_f)t$ |

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Period and Frequency

For uniform circular motion (constant $\omega$):

Period $T$

Time for one complete rotation: $T = \frac{2\pi}{\omega}$

Units: seconds

Frequency $f$

Rotations per second: $f = \frac{1}{T} = \frac{\omega}{2\pi}$

Units: Hz (hertz) = rev/s

Relationships

$\omega = 2\pi f = \frac{2\pi}{T}$

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Rolling Motion

For an object rolling without slipping:

Constraint condition: $v_{cm} = r\omega$

where:

• $v_{cm}$ = velocity of center of mass
• $r$ = radius
• $\omega$ = angular velocity

No slipping means:

• Point of contact is instantaneously at rest
• Distance traveled = arc length: $s = r\theta$

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Problem-Solving Strategy

For Rotational Kinematics:

1. Identify knowns: $\theta_i$, $\omega_i$, $\omega_f$, $\alpha$, $t$, $\Delta\theta$
2. Identify unknown: What are you solving for?
3. Choose equation: Pick the one with known quantities and unknown
4. Solve algebraically
5. Check units: Should be rad, rad/s, or rad/s²
6. Check reasonableness: Does answer make sense?

Tip: If you know linear quantities ($v$, $a_t$, $s$), convert using:

• $\omega = v/r$
• $\alpha = a_t/r$
• $\theta = s/r$

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⚠️ Common Mistakes

Mistake 1: Degrees vs. Radians

MUST use radians in equations! Convert degrees to radians first.

Mistake 2: Confusing $a_t$ and $a_c$

• $a_t = r\alpha$: tangential (changes speed)
• $a_c = \omega^2 r$: centripetal (changes direction)
• Total: $a = \sqrt{a_t^2 + a_c^2}$

Mistake 3: Wrong Sign for $\alpha$

• Speeding up in positive direction: $\alpha > 0$
• Slowing down in positive direction: $\alpha < 0$

Mistake 4: Forgetting Initial Conditions

$\omega_i$ and $\theta_i$ are not always zero!

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Special Cases

Starting from Rest

$\omega_i = 0$:

• $\omega_f = \alpha t$
• $\theta = \frac{1}{2}\alpha t^2$
• $\omega_f^2 = 2\alpha\theta$

Uniform Rotation

$\alpha = 0$ (constant $\omega$):

• $\omega_f = \omega_i = \omega$
• $\theta = \omega t$
• Period: $T = \frac{2\pi}{\omega}$

Coming to Rest

$\omega_f = 0$:

• $0 = \omega_i + \alpha t$ → $t = -\frac{\omega_i}{\alpha}$
• $\omega_i^2 = -2\alpha\Delta\theta$ → $\Delta\theta = -\frac{\omega_i^2}{2\alpha}$

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Applications

Wheels and Gears

• Angular velocity determines linear speed
• Gear ratios change angular velocities
• $v = r\omega$ connects the two

Rotating Machinery

• Turbines, engines, motors
• Angular acceleration during startup
• Constant $\omega$ during normal operation

Sports

• Figure skating spins (angular velocity)
• Gymnastics rotations
• Diving somersaults

Astronomy

• Planetary rotation (Earth: $T \approx 24$ hr)
• Orbital motion
• Galaxy rotation

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Key Formulas Summary

| Quantity | Formula | Units | |----------|---------|-------| | Angular velocity | $\omega = \frac{d\theta}{dt}$ | rad/s | | Angular acceleration | $\alpha = \frac{d\omega}{dt}$ | rad/s² | | Linear velocity | $v = r\omega$ | m/s | | Tangential acceleration | $a_t = r\alpha$ | m/s² | | Centripetal acceleration | $a_c = \omega^2 r$ | m/s² | | Period | $T = \frac{2\pi}{\omega}$ | s | | Frequency | $f = \frac{\omega}{2\pi}$ | Hz |

Kinematic equations (constant $\alpha$):

1. $\omega_f = \omega_i + \alpha t$
2. $\theta = \theta_i + \omega_i t + \frac{1}{2}\alpha t^2$
3. $\omega_f^2 = \omega_i^2 + 2\alpha\Delta\theta$