Loading [MathJax]/jax/output/HTML-CSS/jax.js
Skip to main content
Library homepage
 

Text Color

Text Size

 

Margin Size

 

Font Type

Enable Dyslexic Font
Physics LibreTexts

13.S: Rigid-body Rotation (Summary)

( \newcommand{\kernel}{\mathrm{null}\,}\)

This chapter has introduced the important, topic of rigid-body rotation which has many applications in physics, engineering, sports, etc.

Inertia tensor

The concept of the inertia tensor was introduced where the 9 components of the inertia tensor are given by

Iij=ρ(r)(δij(3kx2k)xixj)dV

Steiner’s parallel-axis theorem

J11I11+M((a21+a22+a33)δ11a21)=I11+M(a22+a23)

relates the inertia tensor about the center-of-mass to that about parallel axis system not through the center of mass.

Diagonalization of the inertia tensor about any point was used to find the corresponding Principal axes of the rigid body.

Angular momentum

The angular momentum L for rigid-body rotation is expressed in terms of the inertia tensor and angular frequency ω by

L=(I11I12I13I21I22I23I31I32I33)(ω1ω2ω3)={I}ω

Rotational kinetic energy

The rotational kinetic energy is

Trot=12(ω1 ω2 ω3)(I11I12I13I21I22I23I31I32I33)(ω1ω2ω3)

TrotT=12ω{I}ω=12ωL

Euler angles

The Euler angles relate the space-fixed and body-fixed principal axes. The angular velocity ω expressed in terms of the Euler angles has components for the angular velocity in the body-fixed axis system (1,2,3)

ω1=˙ϕ1+˙θ1+˙ψ1=˙ϕsinθsinψ+˙θcosψ

ω2=˙ϕ2+˙θ2+˙ψ2=˙ϕsinθcosψ˙θsinψ

ω3=˙ϕ3+˙θ3+˙ψ3=˙ϕcosθ+˙ψ

Similarly, the components of the angular velocity for the space-fixed axis system (x,y,z) are

ωx=˙θcosϕ+˙ψsinθsinϕ

ωy=˙θsinϕ˙ψsinθcosϕ

ωz=˙ϕ+˙ψcosθ

Rotational invariants

The powerful concept of the rotational invariance of scalar properties was introduced. Important examples of rotational invariants are the Hamiltonian, Lagrangian, and Routhian.

Euler equations of motion for rigid-body motion

The dynamics of rigid-body rotational motion was explored and the Euler equations of motion were derived using both Newtonian and Lagrangian mechanics.

Next1=I1˙ω1(I2I3)ω2ω3Next2=I2˙ω2(I3I1)ω3ω1Next3=I3˙ω3(I1I2)ω1ω2

Lagrange equations of motion for rigid-body motion

The Euler equations of motion for rigid-body motion, given in Equation 13.S.12, were derived using the Lagrange-Euler equations.

Torque-free motion of rigid bodies

The Euler equations and Lagrangian mechanics were used to study torque-free rotation of both symmetric and asymmetric bodies including discussion of the stability of torquefree rotation.

Rotating symmetric body subject to a torque

The complicated motion exhibited by a symmetric top, that is spinning about one fixed point and subject to a torque, was introduced and solved using Lagrangian mechanics.

The rolling wheel

The non-holonomic motion of rolling wheels was introduced, as well as the importance of static and dynamic balancing of rotating machinery..

Rotation of deformable bodies

The complicated non-holonomic motion involving rotation of deformable bodies was introduced.


This page titled 13.S: Rigid-body Rotation (Summary) is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Douglas Cline via source content that was edited to the style and standards of the LibreTexts platform.

Support Center

How can we help?