Chapter 12 Summary: Center of Mass, Torque, and Rotational Motion

Degrees of Freedom for Rigid Bodies

  • Translation: 1D objects.

  • Rotation: 2D objects.

  • Translation and Rotation: 2D objects.

  • 3D rigid body has 6 degrees of freedom: Roll, Yaw, Pitch.

Center of Mass (CoM)

  • Definition: "Geometrical average" location of an object's mass.

  • For translational motion, treat the body as a point mass at the CoM.

  • Discrete masses: CoM=∑m<em>ir</em>iMCoM = \frac{\sum m<em>i r</em>i}{M}, where M is the total mass of all particles.

  • Continuous mass: Model as an infinite array of infinitesimally small point masses.

  • Newton’s 2nd Law for an Extended Object: ∑F<em>ext=Ma</em>cm\sum F<em>{ext} = Ma</em>{cm}, where acma_{cm} is the acceleration of the Center of Mass.

Torque

  • Definition: τ=r×F=rFsinθ\tau = r \times F = rFsin\theta- rr: distance from the axis of rotation to the point where the force FF is applied

    • θ\theta: the angle between rr and FF

  • τ=Fr<em>⊥=F</em>⊥r\tau = Fr<em>{\perp} = F</em>{\perp}r (lever arm/perpendicular distance)

Static Equilibrium

  • Translational equilibrium: ∑F=0\sum F = 0

  • Rotational equilibrium: ∑τ=0\sum \tau = 0

  • The net force on the system must be zero.

  • The net torque around the center of mass must be zero.

  • The net torque around any point must be zero.

Rotational Motion

  • Newton's Second Law for Rotation: τ=Iα\tau = I \alpha, where II is the moment of inertia.

  • Linear to Angular conversion using radius (r): ωr=v,αr=a\omega r = v, \quad \alpha r = a

  • ∑τ=Iα\sum \tau = I\alpha – Newton’s 2nd Law for rotations

Relating Linear and Rotational Motion

  • Rim speed = ωR\omega R

  • Rim acceleration = αR\alpha R

  • Object’s motion matches the rim's motion.

  • Nonslipping condition : V<em>obj=ωRV<em>{obj} = \omega R, a</em>obj=aRa</em>{obj} = aR

Moments of Inertia

  • Calculating the Moment of Inertia, divide a solid object into many small cells of mass Δm\Delta m and let Δm→0\Delta m \rightarrow 0.

  • I=∫r2dmI = \int r^2 dm where rr is the distance from the rotation axis.

Kinetic Energy and Rotational Motion

  • Kinetic Energy for an extended rotating mass: KErot=12Iω2KE_{rot} = \frac{1}{2}I\omega^2

Conservation of Energy

  • For extended rigid bodies, consider rotational motion.

  • Additional component to kinetic energy: Rotation KE=12Iω2KE = \frac{1}{2}I\omega^2

  • Law of conservation of energy- KE<em>rot,i+KE</em>tran,i=PE+KE<em>rot,f+KE</em>tran,fKE<em>{rot, i} + KE</em>{tran, i} = PE + KE<em>{rot, f} + KE</em>{tran, f}