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 = \frac{\sum mi ri}{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: \sum F{ext} = Ma{cm}, where a_{cm} is the acceleration of the Center of Mass.

Torque

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

    • \theta: the angle between r and F

  • \tau = Fr{\perp} = F{\perp}r (lever arm/perpendicular distance)

Static Equilibrium

  • Translational equilibrium: \sum F = 0

  • Rotational equilibrium: \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: \tau = I \alpha, where I is the moment of inertia.

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

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

Relating Linear and Rotational Motion

  • Rim speed = \omega R

  • Rim acceleration = \alpha R

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

  • Nonslipping condition : V{obj} = \omega R, a{obj} = aR

Moments of Inertia

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

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

Kinetic Energy and Rotational Motion

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

Conservation of Energy

  • For extended rigid bodies, consider rotational motion.

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

  • Law of conservation of energy- KE{rot, i} + KE{tran, i} = PE + KE{rot, f} + KE{tran, f}