Rotational Motion Formulas

Why This Matters

Rotational motion is everywhere, from car wheels and spinning tops to planets orbiting stars. In College Physics, you're tested on your ability to recognize that rotation follows the same logical structure as linear motion, just with different variables. The formulas here aren't random; they're direct analogs to the $F = ma$ and kinematic equations you already know. If you understand one set, you can master the other.

What separates students who ace rotational motion from those who struggle is understanding the why behind each formula. You need to know when to use torque versus angular momentum, how moment of inertia affects rotational behavior, and why energy and momentum conservation apply to spinning objects just as they do to moving ones. Don't just memorize formulas; know what physical principle each one represents and when to apply it.

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Describing Rotational Motion: The Kinematics

Just like linear motion has displacement, velocity, and acceleration, rotational motion has angular versions of each. These quantities describe how rotation changes over time without worrying about what causes it.

Angular Displacement: $\Delta\theta = \theta_f - \theta_i$

• Measures the total angle an object has rotated through, the rotational equivalent of displacement
• Expressed in radians for physics calculations, though degrees and revolutions work for everyday contexts. One full revolution = $2\pi$ rad = 360°.
• Sign indicates direction: positive typically means counterclockwise, negative means clockwise

Angular Velocity: $\omega = \frac{\Delta\theta}{\Delta t}$

• Describes rotation rate: how fast an object spins, measured in radians per second (rad/s)
• Vector quantity with direction along the axis of rotation (use the right-hand rule: curl your fingers in the direction of rotation, and your thumb points along $\omega$)
• Connects to period via $\omega = \frac{2\pi}{T}$, useful for problems involving complete rotations. You can also write this as $\omega = 2\pi f$, where $f$ is the frequency in Hz.

Angular Acceleration: $\alpha = \frac{\Delta\omega}{\Delta t}$

• Rate of change of angular velocity: tells you if rotation is speeding up or slowing down
• Positive $\alpha$ means angular velocity is increasing in the positive direction; negative means it's decreasing (or increasing in the negative direction)
• Measured in rad/s², the rotational analog to $m/s^2$ in linear motion

Rotational Kinematics Equations

These work exactly like their linear counterparts, just with angular variables swapped in. They apply only when angular acceleration is constant.

• $\omega = \omega_0 + \alpha t$ mirrors $v = v_0 + at$. Use when you need final angular velocity.
• $\theta = \theta_0 + \omega_0 t + \frac{1}{2}\alpha t^2$ mirrors the position equation. Use for angular displacement over time.
• $\omega^2 = \omega_0^2 + 2\alpha(\theta - \theta_0)$ eliminates time. Use when time isn't given or needed.

The strategy is the same as linear kinematics: identify your three known quantities, pick the equation that contains those three plus the one unknown you're solving for.

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Connecting Linear and Rotational Motion

These formulas bridge the gap between how fast something spins and how fast points on that object actually move. This connection is essential for problems involving rolling, orbiting, or any motion along a curved path.

Tangential Velocity: $v = r\omega$

• Converts angular to linear speed: points farther from the axis move faster even at the same $\omega$
• Direction is tangent to the circular path, hence "tangential"
• Critical for rolling motion: a wheel rolling without slipping has $v_{center} = r\omega$

Tangential Acceleration: $a_t = r\alpha$

• Converts angular acceleration to linear acceleration along the direction of motion
• This is separate from centripetal acceleration; $a_t$ changes the speed of a point on the rim, while $a_c$ changes its direction

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

• Always points toward the center: this is what keeps objects moving in circles rather than flying off in a straight line
• Not caused by a special force: it's the result of forces like tension, gravity, or friction acting toward the center
• Two equivalent forms: use $v^2/r$ when you know linear speed, $r\omega^2$ when you know angular velocity

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What Causes Rotation: Forces and Torque

Understanding why things rotate requires torque and moment of inertia. These concepts parallel force and mass in linear motion, forming the foundation of rotational dynamics.

Torque: $\tau = rF\sin\theta$

• Rotational equivalent of force: measures how effectively a force causes rotation about a pivot
• $r$ is the distance from the pivot to the point where the force is applied, and $\theta$ is the angle between the position vector $\vec{r}$ and the force vector $\vec{F}$. The quantity $r\sin\theta$ gives you the perpendicular distance from the force's line of action to the pivot, often called the lever arm.
• Maximum torque occurs when the force is perpendicular to $\vec{r}$ ($\theta = 90°$, so $\sin\theta = 1$). Zero torque when the force points directly toward or away from the pivot ($\theta = 0°$ or $180°$).
• Units are Newton-meters (N·m): same dimensions as energy, but conceptually different (torque is not energy)

Moment of Inertia: $I = \Sigma mr^2$

• Rotational equivalent of mass: represents resistance to changes in rotational motion
• Depends on mass distribution: mass farther from the axis contributes more to $I$. That's why a hoop (all mass at the rim) has a larger $I$ than a solid disk of the same mass and radius.
• Common shapes to know:

Newton's Second Law for Rotation: $\tau_{net} = I\alpha$

• The rotational $F = ma$: net torque equals moment of inertia times angular acceleration
• Use this whenever you need to find angular acceleration from forces, or vice versa
• Works for systems too: sum all torques about a chosen axis to find the net rotational effect

Parallel Axis Theorem: $I = I_{CM} + Md^2$

• Shifts the rotation axis: calculates moment of inertia about any axis parallel to one through the center of mass
• $d$ is the distance between the center-of-mass axis and the new axis
• Always increases $I$: rotating about an off-center axis is always harder than rotating about the center of mass. This is why the thin rod about its end ($\frac{1}{3}ML^2$) is larger than about its center ($\frac{1}{12}ML^2$).

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Energy in Rotational Systems

Rotating objects store energy and can do work, just like moving objects. These formulas let you apply energy conservation to spinning systems.

Rotational Kinetic Energy: $KE_{rot} = \frac{1}{2}I\omega^2$

• Energy stored in rotation, the rotational analog to $\frac{1}{2}mv^2$
• Rolling objects have both: total $KE = \frac{1}{2}mv^2 + \frac{1}{2}I\omega^2$ for translation plus rotation
• Depends on $I$: objects with larger moments of inertia store more rotational energy at the same $\omega$. For rolling objects, this means more of the gravitational PE goes into spinning rather than translating, so they move slower down a ramp.

Work Done by Torque: $W = \tau\theta$

• Rotational work formula, analogous to $W = Fd$ in linear motion
• $\theta$ must be in radians for the units to work out to joules
• Connects to energy change: the net work done by torque equals the change in rotational kinetic energy (work-energy theorem)

Power in Rotational Motion: $P = \tau\omega$

• Rate of rotational work, analogous to $P = Fv$ in linear motion
• Measured in watts: useful for analyzing motors, engines, and rotating machinery
• High power requires both strong torque and fast rotation

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Momentum and Conservation Laws

Angular momentum is the rotational analog of linear momentum, and its conservation is one of the most powerful principles in physics. Use these when external torques are zero or when analyzing collisions and interactions.

Angular Momentum: $L = I\omega$

• Quantity of rotational motion, the rotational equivalent of $p = mv$
• Vector quantity: direction follows the right-hand rule along the rotation axis
• Depends on both $I$ and $\omega$: changing either one changes angular momentum

Angular Impulse: $\tau_{net} \Delta t = \Delta L$

• Rotational analog of impulse ($F\Delta t = \Delta p$): a net torque applied over time changes angular momentum
• This connects Newton's second law for rotation to angular momentum, since $\tau_{net} = I\alpha = I\frac{\Delta\omega}{\Delta t} = \frac{\Delta L}{\Delta t}$

Conservation of Angular Momentum: $L_i = L_f$ (when $\tau_{ext} = 0$)

• No external torque means constant $L$: the system's total angular momentum is conserved
• The classic "ice skater effect": pulling arms in decreases $I$, so $\omega$ must increase to keep $L$ constant. Quantitatively: $I_i\omega_i = I_f\omega_f$
• Essential for collision problems: use when objects interact rotationally without external torques

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Quick Reference Table

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Self-Check Questions

1. Which two formulas would you use together to find the angular acceleration of a disk when a known force is applied at its edge? What information do you need about the disk?

2. A solid sphere and a hollow sphere of equal mass and radius roll down the same ramp. Which reaches the bottom first, and why? (Hint: think about how $I$ affects energy distribution.)

3. Compare $\tau_{net} = I\alpha$ and $F_{net} = ma$. What plays the role of force? What plays the role of mass? When would you use each equation?

4. An ice skater spinning with arms extended pulls her arms in tight. Explain what happens to her angular velocity, moment of inertia, and rotational kinetic energy using the relevant formulas.

5. A rotating system has no external torques, and something changes internally. Which conservation law applies, and which formula would you set up?