Rotational Motion

Rotational Motion

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 Kinematics Concepts

Just as linear kinematics describes motion along a line ( $x, v, a$ ), angular kinematics describes rotation about a fixed axis ( $\theta, \omega, \alpha$ ).

Angular Position ( $\theta$ )

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

\theta = \frac{s}{r}

(Where $s$ is the arc length and $r$ is the radius).

Angular Velocity ( $\omega$ )

The rate of change of angular position. It tells us how fast an object is spinning. The SI unit is rad/s.

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

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

Angular Acceleration ( $\alpha$ )

The rate of change of angular velocity. The SI unit is $\text{rad/s}^2$ .

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

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

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

Constant Acceleration Equations:

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

Important

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.

Note

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. The SI unit is $\text{N}\cdot\text{m}$ .

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

The magnitude is:

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

(Where $d = r\sin\phi$ is the "lever arm" or perpendicular distance from the axis to the line of action of the force).

I

Moment of Inertia ( $I$ ) Concepts

If torque is the rotational analog to force, what is the analog to mass? It is the Moment of Inertia, which measures an object's resistance to changes in its rotation.

Unlike mass, which is a scalar property of the object itself, the moment of inertia depends on how that mass is distributed relative to the specific axis of rotation.

Moment of Inertia ( $I$ )

For a system of discrete point masses ( $m_i$ ) at distances ( $r_i$ ) from the axis of rotation:

I = \sum m_i r_i^2

For a continuous solid body of density $\rho$ , we integrate over the volume:

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

The SI unit is $\text{kg}\cdot\text{m}^2$ .

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

If you know the moment of inertia through the center of mass ( $I_{cm}$ ), you can easily find the moment of inertia around any parallel axis located a distance $d$ away.

I = I_{cm} + Md^2

Note

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 $F=ma$ .

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

Newton's Second Law for Rotation Concepts

The net torque on a rigid body is equal to its moment of inertia multiplied by its angular acceleration.

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 ( $K_{R}$ )

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

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 ( $\omega$ ).

v_{cm} = R \omega

Rolling Without Slipping Concepts

And similarly for acceleration: $a_{cm} = R \alpha$ .

Because it is both translating and rotating, its total kinetic energy is the sum of its translational and rotational kinetic energies: $K_{total} = \frac{1}{2}Mv_{cm}^2 + \frac{1}{2}I_{cm}\omega^2$ .

The work done by a torque $\tau$ rotating an object through an angle $\Delta\theta$ is:

W = \int \tau \, d\theta

Rolling Without Slipping Concepts

The Work-Energy Theorem for rotation states that the net work done by torques equals the change in rotational kinetic energy.

L

Angular Momentum ( $L$ ) Concepts

The rotational analog to linear momentum ( $p=mv$ ) is angular momentum.

Angular Momentum ( $\vec{L}$ )

For a point particle with linear momentum $\vec{p}$ at a position $\vec{r}$ relative to an origin:

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

For a rigid body rotating about a fixed axis of symmetry:

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

The SI unit is $\text{kg}\cdot\text{m}^2/\text{s}$ .

Angular Momentum ( $L$ ) Concepts

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

Important

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).

Note

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.

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Angular Kinematics

Angular Kinematics Concepts

Angular Position (θ\thetaθ)

Angular Velocity (ω\omegaω)

Angular Acceleration (α\alphaα)

The Analogies

The Analogies Concepts

Kinematic Analogies

Important

Note

Rotational Kinematics & Acceleration Vector Simulator

Rotational Dynamics

Rotational Dynamics Concepts

Torque (τ⃗\vec{\tau}τ)

Moment of Inertia (III)

Moment of Inertia (III) Concepts

Moment of Inertia (III)

Common Moments of Inertia

Parallel-Axis Theorem

Note

Moment Of Rotational Inertia Simulator

Newton's Second Law for Rotation

Newton's Second Law for Rotation Concepts

Newton's Second Law for Rotation Concepts

Rotational Energy and Work

Rotational Energy and Work Concepts

Rotational Kinetic Energy (KRK_RKR​)

Rolling Without Slipping

Rolling Without Slipping Concepts

Rolling Without Slipping Concepts

Rolling Without Slipping Concepts

Angular Momentum (LLL)

Angular Momentum (LLL) Concepts

Angular Momentum (L⃗\vec{L}L)

Angular Momentum (LLL) Concepts

Important

Note

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 ( $I$ ) Concepts

Moment of Inertia ( $I$ )

Rotational Kinetic Energy ( $K_{R}$ )

Angular Momentum ( $L$ )

Angular Momentum ( $L$ ) Concepts

Angular Momentum ( $\vec{L}$ )

Angular Momentum ( $L$ ) Concepts