Rotational Motion Formulas

Why This Matters

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

What separates students who ace rotational motion questions 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 principles 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 distance traveled
• Expressed in radians for physics calculations, though degrees and revolutions work for everyday contexts
• 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)
• Connects to period via $\omega = \frac{2\pi}{T}$, useful for problems involving complete rotations

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 decreasing
• Measured in rad/s²—the rotational analog to $m/s^2$ in linear motion

Rotational Kinematics Equations

• $\omega = \omega_0 + \alpha t$ 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

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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 $v_{center} = r\omega$

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
• Not caused by a special force—it's the result of forces like tension, gravity, or friction acting centripetally
• 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 introducing 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
• Depends on lever arm—the perpendicular distance from the force to the axis (that's what $r\sin\theta$ calculates)
• Units are Newton-meters (N·m)—same dimensions as energy but conceptually different

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 $I$
• Different shapes have different formulas—solid sphere: $\frac{2}{5}MR^2$; solid cylinder: $\frac{1}{2}MR^2$; hoop: $MR^2$

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

• The rotational $F = ma$—net torque equals moment of inertia times angular acceleration
• Fundamental dynamics equation—use this whenever you need to find angular acceleration from forces
• Works for systems too—sum all torques about a chosen axis to find 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

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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 energy at the same $\omega$

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

• Rotational work formula—analogous to $W = Fd$ in linear motion
• $\theta$ must be in radians—this ensures the units work out to joules
• Connects to energy change—work done by net torque equals change in rotational kinetic energy

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
• Product of torque and angular velocity—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

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
• Explains the "ice skater effect"—pulling arms in decreases $I$, so $\omega$ must increase to keep $L$ constant
• 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 and contrast $\tau = I\alpha$ and $F = ma$. 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. If an FRQ gives you a rotating system with no external torques and asks about the final state after an internal change, which conservation law applies and which formula would you set up?