Rotational Motion

Rotational Motion

Learning Objectives

Understand the analogies between linear and angular kinematics.
Define and apply concepts of torque and moment of inertia.
State and apply Newton's Second Law for rotational motion.
Calculate rotational kinetic energy and apply work-energy principles to rolling motion.
Define angular momentum and apply the principle of conservation of angular momentum.

Everything we have learned about straight-line (translational) motion has a direct analogy in rotational motion. This is crucial for analyzing spinning gears, turbines, and the stability of structures. While we often model objects as simple point masses moving in straight lines, the real world is filled with spinning, twisting, and rotating bodies. From the massive turbines generating our electricity to the microscopic gears in a watch, rotational motion is everywhere. Fortunately, the mathematical framework we built for linear motion maps perfectly onto rotational motion through a set of elegant analogies.

Angular Kinematics

Angular Position ( $\theta$ )

The angle of a rotating body relative to a reference line. It is measured in radians (rad), where $2\pi \text{ rad} = 360^\circ = 1 \text{ revolution}$ .

Angular Position Equation

Relates angular position, arc length, and radius.

\theta = \frac{s}{r}

Variables

Symbol	Description	Unit
$\theta$	Angular position	rad
$s$	Arc length	m
$r$	Radius	m

Angular Velocity ( $\omega$ )

The rate of change of angular position. It tells us how fast an object is spinning.

Angular Velocity Equations

Average and instantaneous angular velocity.

\omega_{avg} = \frac{\Delta\theta}{\Delta t}

\omega(t) = \frac{d\theta}{dt}

Variables

Symbol	Description	Unit
$\omega$	Angular velocity	rad/s
$\Delta\theta$	Change in angular position	rad
$\Delta t$	Time interval	s

Angular Acceleration ( $\alpha$ )

The rate of change of angular velocity.

Angular Acceleration Equations

Average and instantaneous angular acceleration.

\alpha_{avg} = \frac{\Delta\omega}{\Delta t}

\alpha(t) = \frac{d\omega}{dt} = \frac{d^2\theta}{dt^2}

Variables

Symbol	Description	Unit
$\alpha$	Angular acceleration	$rad/s^2$
$\Delta\omega$	Change in angular velocity	rad/s
$\Delta t$	Time interval	s

The Analogies

The Analogies Concepts

Because the calculus relationships between position, velocity, and acceleration are identical in both linear and rotational regimes, the equations of motion for constant angular acceleration are identical in form to the "Big Four" linear equations.

Kinematic Analogies

Linear Quantity	Rotational Quantity	Relation ( $v = r\omega$ )
Position ( $x$ )	Angle ( $\theta$ )	$s = r\theta$
Velocity ( $v$ )	Angular Velocity ( $\omega$ )	$v = r\omega$
Tangential Accel ( $a_t$ )	Angular Accel ( $\alpha$ )	$a_t = r\alpha$

Constant Acceleration Equations:

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

Right-Hand Rule

The vector direction of $\vec{\omega}$ and $\vec{\alpha}$ is determined by the Right-Hand Rule. Curl the fingers of your right hand in the direction of rotation; your thumb points along the axis of rotation in the direction of the angular velocity vector.

Interactive Simulation

Use this rotational kinematics model to connect $\theta$ , $\omega$ , and $\alpha$ with the constant-acceleration equations.

Interactive Physics Simulation

Rotational Kinematics & Acceleration Vector Simulator

Study the motion of a particle rotating on a disk. Observe how physical tangential and centripetal acceleration vector components scale at radius R.

Initial Velocity (ω0)3.0 rad/s

Angular Acceleration (α)0.8 rad/s²

Tracking Point Radius (R)1.5 m

Time Elapsed (t)4.0 s

Constant Acceleration Formulas

Final Velocity:

\omega = \omega_0 + \alpha t

Displacement:

\theta = \omega_0 t + \frac{1}{2}\alpha t^2

Linear Vector Magnitudes:

a_c = \omega^2 R

a_t = \alpha R

Note that centripetal acceleration increases with the square of speed, while tangential acceleration remains constant under constant $\alpha$.

Angular Velocity (ω)

6.20 rad/s

Angular displacement (θ)

18.40 rad

Centripetal Accel. (ac)

57.66 m/s²

Tangential Accel. (at)

1.20 m/s²

Rotational Dynamics

Rotational Dynamics Concepts

To cause a change in linear motion, we apply a force ( $F$ ). To cause a change in rotational motion, we apply a torque ( $\tau$ ).

Torque ( $\vec{\tau}$ )

The rotational equivalent of force; a measure of how much a force acting on an object causes that object to rotate. It is the cross product of the position vector (from the axis of rotation to the point of force application) and the force vector.

Torque Equation

Vector cross product definition of torque.

\vec{\tau} = \vec{r} \times \vec{F}

Variables

Symbol	Description	Unit
$\vec{\tau}$	Torque vector	$N\cdot m$
$\vec{r}$	Position vector from axis	m
$\vec{F}$	Applied force vector	N

Torque Magnitude Equation

Magnitude of torque using lever arm distance.

|\vec{\tau}| = r F \sin\phi = F d

Variables

Symbol	Description	Unit
$\|\vec{\tau}\|$	Magnitude of torque	$N\cdot m$
$r$	Distance from axis to force application	m
$F$	Magnitude of applied force	N
$\phi$	Angle between position and force vectors	rad
$d$	Lever arm distance	m

I

Moment of Inertia ( $I$ )

The rotational analog to mass, which measures an object's resistance to changes in its rotation. It depends on how the mass is distributed relative to the specific axis of rotation.

Moment of Inertia for Discrete Masses

Calculates moment of inertia for a system of point masses.

I = \sum m_i r_i^2

Variables

Symbol	Description	Unit
$I$	Moment of inertia	$kg\cdot m^2$
$m_i$	Mass of particle i	kg
$r_i$	Distance of particle i from axis	m

Moment of Inertia for Continuous Bodies

Calculates moment of inertia for a continuous solid body.

I = \int r^2 \, dm = \int \rho r^2 \, dV

Variables

Symbol	Description	Unit
$I$	Moment of inertia	$kg\cdot m^2$
$r$	Distance from axis	m
$dm$	Infinitesimal mass element	kg
$\rho$	Density of the body	$kg/m^3$
$dV$	Infinitesimal volume element	$m^3$

Common Moments of Inertia

Solid Cylinder or Disk (axis through center): $I = \frac{1}{2}MR^2$
Hoop or Thin Cylindrical Shell (axis through center): $I = MR^2$
Solid Sphere (axis through center): $I = \frac{2}{5}MR^2$
Thin Rod (axis through center): $I = \frac{1}{12}ML^2$
Thin Rod (axis through end): $I = \frac{1}{3}ML^2$

Parallel-Axis Theorem

Finds moment of inertia around an axis parallel to the center of mass axis.

I = I_{cm} + Md^2

Variables

Symbol	Description	Unit
$I$	Moment of inertia about parallel axis	$kg\cdot m^2$
$I_{cm}$	Moment of inertia about center of mass	$kg\cdot m^2$
$M$	Total mass	kg
$d$	Distance between parallel axes	m

Interactive Simulation

Use this inertia model to compare how mass distribution changes resistance to angular acceleration.

Interactive Physics Simulation

Moment Of Rotational Inertia Simulator

Explore how mass and shape distribution affect resistance to rotation. Apply a tangential force to generate torque and angular acceleration.

Rotational Shape Profile

Mass (M)5 kg

Radius (R)1.2 m

Applied Tangent Force (F)15 N

Inertia Equation for Profile

I = \frac{1}{2} M R^2

A solid cylinder / disk has its mass evenly distributed from the center axle out to the radius R. More mass distributed far from the center yields a larger moment of inertia ($I$).

Moment of Inertia (I)

3.60 kg·m²

Applied Torque (τ = F·R)

18.00 N·m

Angular Accel. (α = τ/I)

5.00 rad/s²

Angular Velocity (ω)

0.00 rad/s

Newton's Second Law for Rotation

Newton's Second Law for Rotation Concepts

With torque and moment of inertia defined, we can state the rotational equivalent of $F=ma$ . The net torque on a rigid body is equal to its moment of inertia multiplied by its angular acceleration.

Newton's Second Law for Rotation

Relates net torque, moment of inertia, and angular acceleration.

\Sigma \vec{\tau} = I \vec{\alpha}

Variables

Symbol	Description	Unit
$\Sigma \vec{\tau}$	Net torque	$N\cdot m$
$I$	Moment of inertia	$kg\cdot m^2$
$\vec{\alpha}$	Angular acceleration vector	$rad/s^2$

Rotational Energy and Work

Rotational Energy and Work Concepts

A spinning object has kinetic energy, even if its center of mass is stationary.

Rotational Kinetic Energy ( $K_R$ )

The kinetic energy possessed by an object due to its rotational motion.

Rotational Kinetic Energy Equation

Calculates kinetic energy from rotational motion.

K_R = \frac{1}{2}I\omega^2

Variables

Symbol	Description	Unit
$K_R$	Rotational kinetic energy	J
$I$	Moment of inertia	$kg\cdot m^2$
$\omega$	Angular velocity	rad/s

Rolling Without Slipping

Rolling Without Slipping Concepts

When a wheel or sphere rolls across a surface without slipping, there is a strict relationship between its translational velocity ( $v_{cm}$ ) and its angular velocity ( $\omega$ ). Because it is both translating and rotating, its total kinetic energy is the sum of its translational and rotational kinetic energies: $K_{total} = \frac{1}{2}Mv_{cm}^2 + \frac{1}{2}I_{cm}\omega^2$ .

Rolling Translational Velocity

Relates center of mass velocity to angular velocity.

v_{cm} = R \omega

Variables

Symbol	Description	Unit
$v_{cm}$	Velocity of the center of mass	m/s
$R$	Radius of the rolling object	m
$\omega$	Angular velocity	rad/s

Work-Energy Theorem for Rotation

The work done by a torque $\tau$ rotating an object through an angle $\Delta\theta$ is equal to the change in rotational kinetic energy.

Work Done by Torque

Calculates work done by applying torque over an angle.

W = \int \tau \, d\theta

Variables

Symbol	Description	Unit
$W$	Work done	J
$\tau$	Torque	$N\cdot m$
$d\theta$	Infinitesimal angular displacement	rad

L

Angular Momentum ( $\vec{L}$ )

The rotational analog to linear momentum ( $p=mv$ ). For a rigid body rotating about a fixed axis of symmetry, it is the product of moment of inertia and angular velocity.

Angular Momentum of a Point Particle

Angular momentum for a point particle relative to an origin.

\vec{L} = \vec{r} \times \vec{p}

Variables

Symbol	Description	Unit
$\vec{L}$	Angular momentum vector	$kg\cdot m^2/s$
$\vec{r}$	Position vector	m
$\vec{p}$	Linear momentum vector	$kg\cdot m/s$

Angular Momentum of a Rigid Body

Angular momentum for a rigid body rotating around a fixed axis.

\vec{L} = I\vec{\omega}

Variables

Symbol	Description	Unit
$\vec{L}$	Angular momentum vector	$kg\cdot m^2/s$
$I$	Moment of inertia	$kg\cdot m^2$
$\vec{\omega}$	Angular velocity vector	rad/s

Conservation of Angular Momentum

Just as $\Sigma F = dp/dt$ , the net torque equals the rate of change of angular momentum: $\Sigma \vec{\tau} = d\vec{L}/dt$ .

Conservation Principle

Conservation of Angular Momentum: If the net external torque on a system is zero ( $\Sigma \vec{\tau}_{ext} = 0$ ), the total angular momentum of the system is conserved ( $\vec{L}_i = \vec{L}_f$ ). This explains why a figure skater spins faster when they pull their arms in (decreasing $I$ must increase $\omega$ to keep $L=I\omega$ constant).

Interactive Simulation

Use this angular momentum model to see why pulling mass inward makes angular speed rise when external torque is negligible.

Interactive Physics Simulation

Angular Momentum Skater Simulator

Pull the rotating masses inward or outward. When friction and external torque are neglected, moment of inertia decreases, causing spin velocity to increase.

Initial Arm Radius (ri)1.30 m

Current Arm Radius (rc)0.70 m

Core Skater Inertia (I_body)2.0 kg·m²

Initial Angular Speed (ωi)2.5 rad/s

Governing Laws

Moment of Inertia

I = I_{body} + 2m \cdot r^2

Conservation of Momentum

L = I_i \cdot \omega_i = I_f \cdot \omega_f

Angular Momentum (L)

17.68 kg·m²/s

Current Angular Speed (ω)

5.09 rad/s

Parallel-Axis Theorem in Practice

Calculating Complex Inertia

The Parallel-Axis Theorem is not just an abstract concept; it is vital for calculating the moment of inertia of complex, composite engineering shapes (like I-beams or rotating eccentric cams) where the axis of rotation does not pass through the center of mass of every individual component.

By breaking a complex shape into simple geometric parts (rectangles, circles), finding their individual moments of inertia about their own centers of mass, and then using the Parallel-Axis Theorem ( $I = I_{cm} + Md^2$ ) to shift those moments to the global axis of rotation, engineers can analyze the rotational dynamics of virtually any rigid body.

Key Takeaways

Rotational kinematics equations ( $\theta, \omega, \alpha$ ) are exactly analogous to linear equations ( $x, v, a$ ).
Torque ( $\vec{\tau} = \vec{r} \times \vec{F}$ ) is the rotational analog to force, causing angular acceleration.
Moment of Inertia ( $I = \sum mr^2$ ) is the rotational analog to mass, representing resistance to angular acceleration. It depends on mass distribution.
Newton's Second Law for rotation is $\Sigma \vec{\tau} = I \vec{\alpha}$ .
A spinning object possesses Rotational Kinetic Energy ( $K_R = \frac{1}{2}I\omega^2$ ).
Angular Momentum ( $\vec{L} = I\vec{\omega}$ ) is conserved if the net external torque is zero.

Previous TopicImpulse and Momentum - Theory & Concepts

Quiz Me

Next TopicEquilibrium and Elasticity - Theory & Concepts

Prev Next

Quiz Me

Learning Objectives

Angular Kinematics

Angular Position (θ\thetaθ)

Angular Position Equation

Angular Velocity (ω\omegaω)

Angular Velocity Equations

Angular Acceleration (α\alphaα)

Angular Acceleration Equations

The Analogies

The Analogies Concepts

Kinematic Analogies

Right-Hand Rule

Interactive Simulation

Rotational Kinematics & Acceleration Vector Simulator

Rotational Dynamics

Rotational Dynamics Concepts

Torque (τ⃗\vec{\tau}τ)

Torque Equation

Torque Magnitude Equation

Moment of Inertia (III)

Moment of Inertia (III)

Moment of Inertia for Discrete Masses

Moment of Inertia for Continuous Bodies

Common Moments of Inertia

Parallel-Axis Theorem

Interactive Simulation

Moment Of Rotational Inertia Simulator

Newton's Second Law for Rotation

Newton's Second Law for Rotation Concepts

Newton's Second Law for Rotation

Rotational Energy and Work

Rotational Energy and Work Concepts

Rotational Kinetic Energy (KRK_RKR​)

Rotational Kinetic Energy Equation

Rolling Without Slipping

Rolling Without Slipping Concepts

Rolling Translational Velocity

Work-Energy Theorem for Rotation

Work Done by Torque

Angular Momentum (LLL)

Angular Momentum (L⃗\vec{L}L)

Angular Momentum of a Point Particle

Angular Momentum of a Rigid Body

Conservation of Angular Momentum

Conservation Principle

Interactive Simulation

Angular Momentum Skater Simulator

Parallel-Axis Theorem in Practice

Calculating Complex Inertia

Angular Position ( $\theta$ )

Angular Velocity ( $\omega$ )

Angular Acceleration ( $\alpha$ )

Torque ( $\vec{\tau}$ )

Moment of Inertia ( $I$ )

Moment of Inertia ( $I$ )

Rotational Kinetic Energy ( $K_R$ )

Angular Momentum ( $L$ )

Angular Momentum ( $\vec{L}$ )