Angular Variables and Linear Relations

Angular Position and Displacement

When an object rotates, every point moves through a curved path called an arc. The angle swept by this motion, measured in radians, tells us how far the object has rotated regardless of its size.

$s = r\theta$

The arc length equals radius times angle in radians, linking linear distance to angular rotation.

$s$=arc length (distance along curved path)(meters (m))

$r$=radius of circular path(meters (m))

$\theta$=angular displacement in radians(radians (rad))

$\theta = 2\pi$ rad

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Complete revolution, arc length = circumference ($2\pi r$)

$\theta = 0^\circ$

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No rotation, arc length = 0

Radian Advantage: One radian is the angle where arc length equals radius, making $s = r\theta$ elegantly simple

Conversion Check: 1 revolution = $2\pi$ rad = 360°, so convert revolutions to radians before using formulas

Angular Displacement: $\Delta\theta = \theta_f - \theta_i$, measured counterclockwise as positive

Full Rotation Significance: Points closer to center travel shorter arcs but sweep same angle

Proportionality: $s \propto r$ at fixed $\theta$, and $s \propto \theta$ at fixed $r$ — doubling either doubles arc length

Angular Velocity (The Rate of Spin)

Angular velocity measures how quickly an object rotates, just like linear velocity measures how fast something moves in a straight line. The faster the spin, the greater the angular velocity.

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

Angular velocity equals angular displacement divided by time, showing how rotation rate connects to geometry.

$\omega$=angular velocity (rate of rotation)(radians per second (rad/s))

$\Delta\theta$=change in angular position(radians (rad))

$\Delta t$=time interval(seconds (s))

$\Delta\theta = 2\pi$ rad in 1s

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$\omega = 2\pi$ rad/s (about 6.28 rad/s)

$\omega > 0$

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Counterclockwise rotation (conventionally positive)

Sign Convention: Positive ω = counterclockwise, negative ω = clockwise by convention

RPM to Rad/s: Multiply revolutions per minute by $\frac{2\pi}{60}$ to convert to rad/s

Period Connection: $\omega = \frac{2\pi}{T}$ where T is the time for one complete rotation

Frequency Link: $\omega = 2\pi f$ where f is frequency in hertz (revolutions per second)

The linear velocity of any point on a rotating object depends on both how fast it spins AND how far from the center it is. Points farther out move faster even though they share the same angular velocity.

$v = r\omega$

Tangential speed equals radius times angular velocity, linking linear and rotational motion.

$v$=linear (tangential) velocity(meters per second (m/s))

$r$=distance from rotation axis(meters (m))

$\omega$=angular velocity(radians per second (rad/s))

$r = 0$ (at center)

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$v = 0$ despite any ω - the center doesn't move

$\omega$ doubles

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$v$ doubles - speed is directly proportional to spin rate

Tangential Direction: Velocity is always perpendicular to radius, tangent to the circular path

CD Example: Outer edge of CD (r=6cm) moves 4× faster than inner point (r=1.5cm)

Practical Insight: This is why ceiling fan blades move air faster at their tips than near center

Proportionality: $v \propto r$ at constant $\omega$ — double the radius, double the speed

The direction of angular velocity follows the right-hand rule, creating a vector perpendicular to the plane of rotation. This points along the axis of spin.

$\vec{\omega} = \omega \hat{n}$

Angular velocity as a vector, with direction given by the right-hand rule along the rotation axis.

$\vec{\omega}$=angular velocity vector(rad/s with direction)

$\hat{n}$=unit vector along rotation axis(dimensionless)

$\omega$=magnitude of angular velocity(rad/s)

Counterclockwise rotation

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Vector points toward you (out of page)

Clockwise rotation

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Vector points away from you (into page)

Right-Hand Rule: Curl fingers in rotation direction, thumb points along ω vector

Axis Alignment: ω always points along rotation axis, not in direction of motion

Why This Matters: Essential for torque, angular momentum, and rotational dynamics

Angular Acceleration

When an object's rotation speeds up or slows down, it experiences angular acceleration. This tells us how quickly the angular velocity changes over time.

$\alpha = \frac{\Delta\omega}{\Delta t}$

Angular acceleration equals change in angular velocity divided by time interval.

$\alpha$=angular acceleration(radians per second squared (rad/s²))

$\Delta\omega$=change in angular velocity(radians per second (rad/s))

$\Delta t$=time interval(seconds (s))

$\alpha > 0$

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Angular velocity increasing (speeding up in rotation)

$\alpha < 0$

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Angular velocity decreasing (slowing down)

Speeding Up: ω and α have same sign (both positive or both negative)

Slowing Down: ω and α have opposite signs

Sign Meaning: Positive α means counterclockwise increase OR clockwise decrease (more negative)

Angular acceleration causes tangential acceleration, which changes the speed of points moving along their circular paths. But rotating objects also have centripetal acceleration even at constant speed.

$a_t = r\alpha, \quad a_c = \omega^2 r$

Tangential acceleration from speeding rotation, centripetal acceleration from curved path direction change.

$a_t$=tangential acceleration (speed change)(m/s²)

$a_c$=centripetal acceleration (direction change)(m/s²)

$\alpha$=angular acceleration(rad/s²)

$\omega$=angular velocity(rad/s)

$\alpha = 0$ (constant ω)

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$a_t = 0$ but $a_c \neq 0$ - still has centripetal acceleration

$\omega = 0$ (starting from rest)

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$a_c = 0$ - only tangential acceleration exists

Tangential Direction: Points along tangent, either same direction as v (speeding up) or opposite (slowing)

Centripetal Direction: Always points toward center of circle, perpendicular to tangential

Total Acceleration: Since $\vec{a}_t \perp \vec{a}_c$, the magnitude is $a = \sqrt{a_t^2 + a_c^2}$

Proportionality: $a_c \propto \omega^2$ — doubling angular velocity quadruples centripetal acceleration

Car Turn Example: Steering wheel angle gives centripetal, gas/brake gives tangential

In uniform circular motion, the angular velocity remains constant. While the object continuously accelerates toward the center, it doesn't speed up or slow down in its rotation.

$\alpha = 0, \quad \omega = \text{constant}$

Uniform circular motion has zero angular acceleration but non-zero centripetal acceleration.

$\alpha = 0$=no change in rotation rate(rad/s²)

$a_c = \omega^2 r$=constant centripetal acceleration(m/s²)

Satellite in orbit

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Uniform circular motion with constant ω

CD spinning at constant speed

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$\alpha = 0$ but edge points have $a_c$

Key Insight: Speed doesn't change, but direction does - that's why there's acceleration

Force Requirement: Centripetal force needed to maintain circular path

Energy Conservation: In uniform circular motion, kinetic energy stays constant

The Equations of Rotational Motion

The kinematic analogy between linear and angular motion is perfect - every linear equation has an angular counterpart. When angular acceleration is constant, we can predict rotational motion precisely.

$\omega_f^2 = \omega_i^2 + 2\alpha\Delta\theta$

The 'timeless' rotational kinematic equation, useful when time isn't given or needed.

$\omega_f$=final angular velocity(rad/s)

$\omega_i$=initial angular velocity(rad/s)

$\alpha$=constant angular acceleration(rad/s²)

$\Delta\theta$=angular displacement(rad)

$\omega_i = 0$ (from rest)

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$\omega_f = \sqrt{2\alpha\Delta\theta}$ - simpler form

$\omega_f = 0$ (to rest)

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$\Delta\theta = -\frac{\omega_i^2}{2\alpha}$ - stopping distance

Linear-Angular Map: x→θ, v→ω, a→α, t→t (time stays the same!)

Other Equations: $\omega_f = \omega_i + \alpha t$ and $\Delta\theta = \omega_i t + \frac{1}{2}\alpha t^2$

When to Use: This equation is perfect when you know ω values, α, and need Δθ (or vice versa)

The complete set of rotational kinematic equations mirrors the linear ones perfectly. All require constant angular acceleration to be valid.

$$\begin{aligned}\omega_f &= \omega_i + \alpha t \\\\ \Delta\theta &= \omega_i t + \frac{1}{2}\alpha t^2 \\\\ \Delta\theta &= \left(\frac{\omega_i + \omega_f}{2}\right)t \\\\ \omega_f^2 &= \omega_i^2 + 2\alpha\Delta\theta\end{aligned}$$

Four rotational kinematic equations for constant angular acceleration, analogous to linear kinematics.

$t$=time interval(seconds (s))

$\frac{\omega_i + \omega_f}{2}$=average angular velocity(rad/s)

$\alpha = 0$

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Reduces to $\omega_f = \omega_i$ and $\Delta\theta = \omega t$

Starting from rest

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$\omega_i = 0$, simplifying all equations

Equation 1: Relates velocities, acceleration, and time (no θ needed)

Equation 2: Relates displacement, time, and acceleration (no ω_f needed)

Equation 3: Relates displacement, average velocity, and time (no α needed)

Equation 4: Relates velocities, acceleration, and displacement (no t needed)

Strategy: List knowns, identify unknown, choose equation with all knowns + unknown