Chapter 12: Rotation of a Rigid Body Notes

12.1 Rotational Motion

• Rigid body: size and shape remain constant during movement.

• Rigid bodies can undergo translational, rotational, or combined motion.

12.2 Rotation About the Center of Mass

• Center of mass: mass-weighted center of an object.

• For symmetrical objects with uniform density, the center of mass is at the physical center.

• Center of mass equations:

  • $x<em>{cm} = \frac{1}{M} \sum</em>{i} m<em>{i}x</em>{i} = \frac{m<em>1x</em>1 + m<em>2x</em>2 + m<em>3x</em>3 + \cdots}{m<em>1 + m</em>2 + m_3 + \cdots}$

  • $y<em>{cm} = \frac{1}{M} \sum</em>{i} m<em>{i}y</em>{i} = \frac{m<em>1y</em>1 + m<em>2y</em>2 + m<em>3y</em>3 + \cdots}{m<em>1 + m</em>2 + m_3 + \cdots}$

• Unconstrained objects with no net force rotate about their center of mass.

12.3 Rotational Energy

• Rotational kinetic energy: $K_{rot} = \frac{1}{2}I\omega^2$

• Mechanical energy of a system with an object rotating about a stationary axle: $E<em>{mech} = K</em>{rot} + U<em>G = \frac{1}{2}I\omega^2 + Mgy</em>{cm}$

12.4 Calculating Moment of Inertia

• Moment of inertia (I): measures an object's resistance to changes in rotational motion.

  • Depends on axis location and mass distribution.

  • $I = \sum<em>{i} m</em>{i}r_{i}^2$ for discrete particles.

  • $I = \int r^2 dm$ for continuous objects.

• Parallel-axis theorem: $I = I_{cm} + Md^2$, where $d$ is the distance from the axis of interest to a parallel axis through the center of mass.

12.5 Torque

• Torque ($\tau$): rotational equivalent of force, measures the effectiveness of a force to cause rotation.

  • $\tau = rF \sin{\phi} = rF_t = Fd$

  • SI units: Newton-meters (Nm).

• Positive torque: counterclockwise rotation.

• Negative torque: clockwise rotation.

• Net torque: $\tau<em>{net} = \sum</em>{i} \tau_i$

• Torque due to gravity: $\tau<em>{grav} = -Mgx</em>{cm}$

12.6 Rotational Dynamics

• Newton’s second law for rotational motion: $\alpha = \frac{\tau_{net}}{I}$

12.7 Rotation About a Fixed Axis

• Ropes and Pulleys

12.8 Static Equilibrium

• Static equilibrium: object is stationary with no net force and no net torque.

  • $\sum \vec{F}_i = 0$

  • $\sum \tau_i = 0$

• Critical Angle: $\theta_c = tan^{-1} {\frac{t}{2h}}$

  • $t$ is track width, $h$ is the the height of center of mass

12.9 Rolling Motion

• Rolling without slipping condition: $v_{cm} = R\omega$

• Kinetic energy of a rolling object: $K<em>{rolling} = K</em>{rot} + K<em>{cm} = \frac{1}{2}I</em>{cm}\omega^2 + \frac{1}{2}Mv_{cm}^2$

12.10 The Vector Description of Rotational Motion

• Angular velocity vector ($\vec{\omega}$): magnitude is angular speed, direction is along the axis of rotation (right-hand rule).

• Torque vector: $\vec{\tau} = \vec{r} \times \vec{F}$

12.11 Angular Momentum

• Angular momentum of a particle: $\vec{L} = \vec{r} \times \vec{p} = mr \nu \sin{\beta}$

• Angular momentum for an extended object: $L = I\omega$

• Conservation Laws: If the net external torque on a system is zero, the total angular momentum is conserved.

12.12 Precession of a Gyroscope