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Physics C: Mechanics Unit 5: Rotational Kinematics and Moment of Inertia

Rotation has its own versions of position, velocity, acceleration and mass. This unit builds the angular variables and the moment of inertia, then uses rotational kinetic energy to analyze spinning and rolling objects.

Covers: Angular position, velocity and acceleration, Constant angular acceleration, Linking linear and angular quantities, Moment of inertia by summation and integration, Parallel-axis theorem, Rotational kinetic energy, Rolling without slipping (energy)

Study guide

A rigid body rotating about a fixed axis is described by its angular position θ(t), measured in radians. Angular velocity is ω = dθ/dt and angular acceleration is α = dω/dt, exactly like the linear case. When α is constant, the equations carry over directly: ω = ω₀ + αt, θ = θ₀ + ω₀t + ½αt² and ω² = ω₀² + 2αΔθ. When α varies, integrate. Watch units: 1 revolution = 2π rad, so rpm must be multiplied by 2π/60 to give rad/s. Every point on a rigid body shares the same ω and α.

Points at different distances from the axis move differently. A point at radius r has tangential speed v = ωr and tangential acceleration a_t = αr, which changes its speed. It also has centripetal acceleration a_c = ω²r = v²/r toward the axis, which changes its direction. These two components are perpendicular, so the total acceleration is √(a_t² + a_c²). Even at constant ω, a point on a spinning wheel accelerates because its direction keeps changing.

The moment of inertia I measures how hard it is to change an object's rotation. For point masses, I = Σmᵢrᵢ², and for continuous bodies, I = ∫r² dm, where r is the perpendicular distance from the axis. Because r is squared, mass far from the axis dominates: a hoop has I = MR², a solid disk ½MR², a solid sphere ⅖MR², a thin spherical shell ⅔MR², and a rod ML²/12 about its center or ML²/3 about its end. For non-uniform bodies, write dm = λ dx (or σ dA) and integrate. Moments of inertia about the same axis add, and a removed hole can be subtracted.

The parallel-axis theorem, I = I_cm + Md², gives I about any axis parallel to one through the center of mass, where d is the distance between the axes. It explains why a rod about its end (ML²/3) is harder to spin than about its center (ML²/12): ML²/12 + M(L/2)² = ML²/3. The center-of-mass axis always gives the smallest I among a family of parallel axes. For flat plates, the perpendicular-axis theorem I_z = I_x + I_y can also help.

A rotating body has kinetic energy ½Iω². An object rolling without slipping has ω = v/R, and its total kinetic energy is ½mv² + ½Iω² = ½mv²(1 + β), where β = I/(mR²). When objects roll from rest down the same ramp, mgh = ½mv²(1 + β) gives v = √(2gh/(1 + β)), independent of mass and radius. Objects with smaller β win: a solid sphere beats a solid cylinder, which beats a hoop. Rolling without slipping involves static friction, which does no work, so mechanical energy is conserved.

Key ideas

• Angular variables θ, ω and α are related by derivatives just like x, v and a.
• Every point on a rigid body has the same ω and α, but linear speed grows with r.
• Total acceleration of a point combines perpendicular tangential and centripetal parts.
• Moment of inertia depends on mass distribution relative to the axis, weighted by r².
• The parallel-axis theorem shifts I from a center-of-mass axis to any parallel axis.
• Rotational kinetic energy is ½Iω².
• Rolling objects share energy between translation and rotation, so lower β means faster.

Formulas and rules

Common mistakes

• Leaving angles in revolutions or degrees when using rotational formulas.
• Measuring d in the parallel-axis theorem from the end of the object instead of from the center of mass.
• Using a uniform-body formula like ML²/3 for an object with non-uniform density.
• Adding tangential and centripetal accelerations as scalars.
• Forgetting rotational kinetic energy for a rolling object.

Before the test, make sure you can

• Differentiate and integrate θ(t), ω(t) and α(t).
• Solve constant angular acceleration problems, including converting to revolutions.
• Convert between angular and linear quantities and find total acceleration.
• Calculate I for point masses, standard shapes and non-uniform rods by integration.
• Apply the parallel-axis theorem correctly.
• Calculate rotational kinetic energy for spinning bodies.
• Use energy conservation to find speeds of rolling objects and rank them.

Words to know

Radian

The angle that subtends an arc equal to the radius; 2π rad = 1 revolution.

Angular velocity

The rate of change of angular position, ω = dθ/dt.

Angular acceleration

The rate of change of angular velocity, α = dω/dt.

Tangential speed

The linear speed of a point on a rotating body, v = ωr.

Tangential acceleration

The acceleration along the path that changes speed, a_t = αr.

Centripetal acceleration

The acceleration toward the axis that changes direction, a_c = ω²r.

Rigid body

An object whose parts keep fixed distances from each other.

Moment of inertia

Rotational inertia, I = ∫r² dm, about a specified axis.

Parallel-axis theorem

I = I_cm + Md² for an axis parallel to a center-of-mass axis at distance d.

Perpendicular-axis theorem

For a flat plate, I_z = I_x + I_y.

Rotational kinetic energy

Energy of rotation, ½Iω².

Rolling without slipping

Rolling in which the contact point is momentarily at rest, so v = ωR.

Shape factor β

The ratio I/(mR²), which sets how a rolling object shares its energy.

Radius of gyration

The distance k at which all the mass could sit to give the same I, so I = Mk².

Sample practice questions

30 more questions like these, from level 1 to 5, are in the practice set.