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,ax, v, a), angular kinematics describes rotation about a fixed axis (θ,ω,α\theta, \omega, \alpha).

Angular Position (θ\theta)

The angle of a rotating body relative to a reference line. It is measured in radians (rad), where 2π rad=360∘=1 revolution2\pi \text{ rad} = 360^\circ = 1 \text{ revolution}.
θ=sr \theta = \frac{s}{r}
(Where ss is the arc length and rr is the radius).

Angular Velocity (ω\omega)

The rate of change of angular position. It tells us how fast an object is spinning. The SI unit is rad/s.
ωavg=ΔθΔt \omega_{avg} = \frac{\Delta\theta}{\Delta t} ω(t)=dθdt \omega(t) = \frac{d\theta}{dt}

Angular Acceleration (α\alpha)

The rate of change of angular velocity. The SI unit is rad/s2\text{rad/s}^2.
αavg=ΔωΔt \alpha_{avg} = \frac{\Delta\omega}{\Delta t} α(t)=dωdt=d2θdt2 \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

| Linear Quantity | Rotational Quantity | Relation (v=rωv = r\omega) | | :--- | :--- | :--- | | Position (xx) | Angle (θ\theta) | s=rθs = r\theta | | Velocity (vv) | Angular Velocity (ω\omega) | v=rωv = r\omega | | Tangential Accel (ata_t) | Angular Accel (α\alpha) | at=rαa_t = r\alpha |
Constant Acceleration Equations:
  • ωf=ωi+αt\omega_f = \omega_i + \alpha t
  • Δθ=ωit+12αt2\Delta\theta = \omega_i t + \frac{1}{2}\alpha t^2
  • ωf2=ωi2+2αΔθ\omega_f^2 = \omega_i^2 + 2\alpha\Delta\theta
  • Δθ=(ωi+ωf2)t\Delta\theta = \left(\frac{\omega_i + \omega_f}{2}\right)t

Important

The vector direction of ω⃗\vec{\omega} and α⃗\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.

Rotational Dynamics

Rotational Dynamics Concepts

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

Torque (τ⃗\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 N⋅m\text{N}\cdot\text{m}.
τ⃗=r⃗×F⃗ \vec{\tau} = \vec{r} \times \vec{F}
The magnitude is:
∣τ⃗∣=rFsin⁡ϕ=Fd |\vec{\tau}| = r F \sin\phi = F d
(Where d=rsin⁡ϕd = r\sin\phi is the "lever arm" or perpendicular distance from the axis to the line of action of the force).

Moment of Inertia (II)

Moment of Inertia (II) 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 (II)

For a system of discrete point masses (mim_i) at distances (rir_i) from the axis of rotation:
I=∑miri2 I = \sum m_i r_i^2
For a continuous solid body of density ρ\rho, we integrate over the volume:
I=∫r2 dm=∫ρr2 dV I = \int r^2 \, dm = \int \rho r^2 \, dV
The SI unit is kg⋅m2\text{kg}\cdot\text{m}^2.

Common Moments of Inertia

  • Solid Cylinder or Disk (axis through center): I=12MR2I = \frac{1}{2}MR^2
  • Hoop or Thin Cylindrical Shell (axis through center): I=MR2I = MR^2
  • Solid Sphere (axis through center): I=25MR2I = \frac{2}{5}MR^2
  • Thin Rod (axis through center): I=112ML2I = \frac{1}{12}ML^2
  • Thin Rod (axis through end): I=13ML2I = \frac{1}{3}ML^2

Parallel-Axis Theorem

If you know the moment of inertia through the center of mass (IcmI_{cm}), you can easily find the moment of inertia around any parallel axis located a distance dd away.
I=Icm+Md2 I = I_{cm} + Md^2

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=maF=ma.
Στ⃗=Iα⃗ \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 (KRK_R)

KR=12Iω2 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 (vcmv_{cm}) and its angular velocity (ω\omega).
vcm=Rω v_{cm} = R \omega

Rolling Without Slipping Concepts

And similarly for acceleration: acm=Rαa_{cm} = R \alpha.
Because it is both translating and rotating, its total kinetic energy is the sum of its translational and rotational kinetic energies: Ktotal=12Mvcm2+12Icmω2K_{total} = \frac{1}{2}Mv_{cm}^2 + \frac{1}{2}I_{cm}\omega^2.
The work done by a torque τ\tau rotating an object through an angle Δθ\Delta\theta is:
W=∫τ dθ 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.

Angular Momentum (LL)

Angular Momentum (LL) Concepts

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

Angular Momentum (L⃗\vec{L})

For a point particle with linear momentum p⃗\vec{p} at a position r⃗\vec{r} relative to an origin:
L⃗=r⃗×p⃗ \vec{L} = \vec{r} \times \vec{p}
For a rigid body rotating about a fixed axis of symmetry:
L⃗=Iω⃗ \vec{L} = I\vec{\omega}
The SI unit is kg⋅m2/s\text{kg}\cdot\text{m}^2/\text{s}.

Angular Momentum (LL) Concepts

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

Important

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

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=Icm+Md2I = 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 (θ,ω,α\theta, \omega, \alpha) are exactly analogous to linear equations (x,v,ax, v, a).
  • Torque (τ⃗=r⃗×F⃗\vec{\tau} = \vec{r} \times \vec{F}) is the rotational analog to force, causing angular acceleration.
  • Moment of Inertia (I=∑mr2I = \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 Στ⃗=Iα⃗\Sigma \vec{\tau} = I \vec{\alpha}.
  • A spinning object possesses Rotational Kinetic Energy (KR=12Iω2K_R = \frac{1}{2}I\omega^2).
  • Angular Momentum (L⃗=Iω⃗\vec{L} = I\vec{\omega}) is conserved if the net external torque is zero.
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