§10.1 — Rotational Kinematics

So far we've studied linear motion — an object moving along a line or in a plane. But many objects rotate: wheels, electrons in atoms, planets, even galaxies. Rotational kinematics rewrites the rules of linear motion for spinning — but with angular quantities instead of linear ones.

Linear and Angular Parallels

There is a deep parallel between the two types of motion:

Linear	Angular	Symbol
Position	Angle	\( \theta \)
Velocity	Angular velocity	\( \omega \)
Acceleration	Angular acceleration	\( \alpha \)

Angle (\( \theta \))

An angle measures the amount of rotation. In physics, we use radians (not degrees).

Definition of Radian

One radian is the angle subtended by an arc whose length equals the radius:

\[ \theta = \frac{s}{r} \]

where:

• \( s \) = arc length
• \( r \) = radius
• \( \theta \) = angle in radians

Example: A circle of radius 2 m. If an object travels along the circumference 6.28 m (≈ \( 2\pi \) meters), what angle is swept?

\[ \theta = \frac{6.28}{2} \approx 3.14 = \pi \text{ rad} \]

This is half a revolution (\( \pi \) radians).

Converting Radians and Degrees

\[ 1 \text{ rev} = 2\pi \text{ rad} = 360° \]

\[ \theta_{\text{rad}} = \theta_{\text{deg}} \times \frac{\pi}{180} \]

Important: In all physics formulas, angles must be in radians, or results will be wrong.

Angular Velocity (\( \omega \))

Angular velocity measures how fast something rotates.

Definition

\[ \omega = \frac{d\theta}{dt} \]

Unit: radians per second (rad/s)

Angular and Linear Velocity

If an object rotates on a circle of radius \( r \) with angular velocity \( \omega \), what is its linear (tangential) speed?

\[ v = \omega r \]

Proof: Arc length \( s = r\theta \). Differentiate:

\[ \frac{ds}{dt} = r\frac{d\theta}{dt} \implies v = \omega r \]

Example: An object rotates on a circle of radius 0.5 m with angular velocity \( \omega = 4 \) rad/s. What is its linear speed?

\[ v = 4 \times 0.5 = 2 \text{ m/s} \]

Angular Acceleration (\( \alpha \))

Angular acceleration measures how fast the angular velocity changes.

Definition

\[ \alpha = \frac{d\omega}{dt} = \frac{d^2\theta}{dt^2} \]

Unit: radians per second squared (rad/s²)

Angular and Tangential Acceleration

If an object rotates on a circle of radius \( r \) with angular acceleration \( \alpha \), what is its tangential (linear) acceleration?

\[ a_{\text{tangential}} = \alpha r \]

Note: This is tangential acceleration (parallel to velocity). The centripetal acceleration \( a_c = \omega^2 r \) is perpendicular to velocity, pointing toward the center.

Example: A wheel starts from rest and accelerates uniformly at \( \alpha = 2 \) rad/s² for 5 seconds. What are its final angular velocity and total angle?

\[ \omega_f = \omega_0 + \alpha t = 0 + 2 \times 5 = 10 \text{ rad/s} \]

\[ \theta = \omega_0 t + \frac{1}{2}\alpha t^2 = 0 + \frac{1}{2} \times 2 \times 25 = 25 \text{ rad} \]

Number of revolutions: \( 25 / (2\pi) \approx 3.98 \) revolutions

Rotational Kinematic Equations

When angular acceleration is constant, the linear kinematic equations translate directly:

Linear (constant \( a \))	Angular (constant \( \alpha \))
\( v = v_0 + at \)	\( \omega = \omega_0 + \alpha t \)
\( x = x_0 + v_0 t + \frac{1}{2}at^2 \)	\( \theta = \theta_0 + \omega_0 t + \frac{1}{2}\alpha t^2 \)
\( v^2 = v_0^2 + 2a(x-x_0) \)	\( \omega^2 = \omega_0^2 + 2\alpha(\theta-\theta_0) \)

Applied example: A stationary wheel begins to rotate. After 3 seconds, its angular velocity is 12 rad/s. Assuming constant angular acceleration, what is \( \alpha \)?

\[ \omega = \omega_0 + \alpha t \implies 12 = 0 + \alpha \times 3 \implies \alpha = 4 \text{ rad/s}^2 \]

Period and Frequency in Rotational Motion

• Period (\( T \)): time for one complete revolution (seconds)
• Frequency (\( f \)): number of revolutions per second (hertz, Hz)

\[ f = \frac{1}{T},\quad \omega = 2\pi f = \frac{2\pi}{T} \]

Example: A ball completes one spin in 0.4 seconds. What are its frequency and angular velocity?

\[ f = \frac{1}{0.4} = 2.5 \text{ Hz} \]

\[ \omega = 2\pi \times 2.5 = 5\pi \approx 15.7 \text{ rad/s} \]

Angular Quantities as Vectors

Although we've focused on magnitudes here, \( \omega \) and \( \alpha \) are actually vectors.

• Direction of \( \vec{\omega} \): Given by the right-hand rule — fingers point in rotation direction, thumb points along \( \vec{\omega} \)
• Direction of \( \vec{\alpha} \): Parallel to \( \vec{\omega} \) if rotation is speeding up; opposite if slowing down

In later chapters, these vectors are crucial for describing torque and angular momentum.

What You Should Know

• Radian definition: \( \theta = s/r \)
• Linear-angular relations: \( v = \omega r \) and \( a_\text{tan} = \alpha r \)
• Kinematic equations: for constant \( \alpha \)
• Counting revolutions: divide radians by \( 2\pi \)
• Period and frequency: \( T = 1/f \), \( \omega = 2\pi/T \)

Preview of §10.2

Now that we know how to describe rotation, the next question: what causes rotation? The answer: torque — the rotational version of force. But to understand torque's effect, we first need moment of inertia — how mass is distributed around the rotation axis.

📚 See also: Halliday Vol 1, Ch 10, §10.1 — Rotational kinematics. 🔗 Reference: §4.4 (Uniform circular motion) — where angular velocity was first introduced.

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