AcademyRigid-Body Rotation
Academy
Connecting Linear and Angular Motion
Level 1 - Physics topic page in Rigid-Body Rotation.
Principle
For a rigid body, linear motion at a point scales with distance from the axis.
Notation
\(r\)
distance from rotation axis
\(\mathrm{m}\)
\(s\)
arc length
\(\mathrm{m}\)
\(v_t\)
tangential speed
\(\mathrm{m\,s^{-1}}\)
\(a_t\)
tangential acceleration
\(\mathrm{m\,s^{-2}}\)
\(a_c\)
centripetal acceleration
\(\mathrm{m\,s^{-2}}\)
\(R\)
rolling radius
\(\mathrm{m}\)
Method
Derivation 1: Start with arc length
Radians are defined so that angular displacement times radius gives arc length. This is why angular formulas connect to linear formulas through \(r\).
Arc length
\[s=r\theta\]
Differentiate once
\[v_t=\frac{ds}{dt}=r\frac{d\theta}{dt}=r\omega\]
Differentiate again
\[a_t=\frac{dv_t}{dt}=r\frac{d\omega}{dt}=r\alpha\]
Derivation 2: Separate tangential and radial acceleration
Tangential acceleration changes speed along the path. Centripetal acceleration changes direction and points inward.
Tangential part
\[a_t=r\alpha\]
Radial part
\[a_c=r\omega^2\]
Using tangential speed
\[a_c=\frac{v_t^2}{r}\]
Derivation 3: Apply the no-slip rolling constraint
Rolling without slipping means the surface contact does not slide. The center advances by one circumference for each full revolution.
No-slip displacement
\[\Delta x_{\mathrm{cm}}=R\Delta\theta\]
No-slip speed
\[v_{\mathrm{cm}}=R\omega\]
No-slip acceleration
\[a_{\mathrm{cm}}=R\alpha\]
Rules
These are the compact results from the method above.
Arc length
\[s=r\theta\]
Tangential speed
\[v_t=r\omega\]
Tangential acceleration
\[a_t=r\alpha\]
Centripetal acceleration
\[a_c=r\omega^2=\frac{v_t^2}{r}\]
Rolling constraint
\[v_{\mathrm{cm}}=R\omega,\qquad a_{\mathrm{cm}}=R\alpha\]
Examples
Question
A point
\[0.30\,\mathrm{m}\]
from an axis has \[\omega=8.0\,\mathrm{rad\,s^{-1}}\]
Find tangential speed.Answer
\[v_t=r\omega=0.30(8.0)=2.4\,\mathrm{m\,s^{-1}}\]
Checks
- Points farther from the axis have larger tangential speed.
- \(a_t\) changes speed; \(a_c\) changes direction.
- Rolling without slipping is a constraint, not an automatic property.
- The contact point is instantaneously at rest only relative to the surface.