Rotation and Rolling

Angular Kinematics

• Angular Variables:
  • Radians: $\theta$
  • Displacement: $\Delta\theta$
  • Velocity: $\omega$
  • Acceleration: $\alpha$

Constant Acceleration

For an object undergoing constant angular acceleration, the linear accelerations are:

• Tangential acceleration: $a_t = \alpha r$
• Centripetal acceleration: $a_n = \frac{v^2}{r} = \omega^2 r$

Rotational Dynamics

Kinetic Energy and Rotational Inertia

The kinetic energy of a rotating rigid body is derived by summing the kinetic energies of its constituent particles:

$$K = \sum \frac{1}{2} m_i (r_i \omega)^2 = \frac{1}{2} \left( \sum m_i r_i^2 \right) \omega^2 = \frac{1}{2} I \omega^2$$

Rotational Inertia (Moment of Inertia):

$$I = \sum m_i r_i^2$$

The Parallel-Axis Theorem

For a rigid body of total mass $M$ rotating about an axis parallel to an axis passing through its center of mass (COM) at a distance $h$:

$$I = I_{\text{com}} + Mh^2$$

Rolling Motion

Rolling motion can be decomposed as a combination of Translation and Rotation.

• Velocity of center of mass: $v_{\text{com}} = \omega R$
• Acceleration of center of mass: $a_{\text{com}} = \alpha R$

For Smooth Rolling, the velocity of a particle $i$ on the object is:

$$v_i = v_{\text{com}} + (r_i \omega) \hat{\theta}_i$$

The total kinetic energy of a rolling object is the sum of its translational and rotational kinetic energies:

$$K = \frac{1}{2} I_{\text{com}} \omega^2 + \frac{1}{2} M v_{\text{com}}^2$$

Rolling on a Ramp

For an object rolling down an incline of angle $\theta$, the acceleration of its center of mass is:

$$a_{\text{com}} = - \frac{g \sin \theta}{1 + \frac{I_{\text{com}}}{m R^2}}$$

Torque

Torque is the rotational equivalent of force on an object.

$$\vec{\tau} = \vec{r} \times \vec{F} = r F \sin \phi$$

• Moment arm: $\vec{r}$
• Line of action: the line along the force vector $\vec{F}$

Newton’s Second Law for Rotation

The net torque on a system is equal to the rate of change of its angular momentum:

$$\vec{\tau}_{\text{net}} = \frac{d\vec{L}}{dt} = I \alpha$$

This relation holds when $I$ is a constant.

Angular Momentum

• For a single particle:

  $$\vec{l} = \vec{r} \times \vec{p} = m (\vec{r} \times \vec{v})$$

• For a system:

  $$\vec{L} = \sum \vec{l} = I \omega$$

Work, Energy, and Power in Rotation

• Work:

  $$W = \int_{\theta_i}^{\theta_f} \tau \, d\theta = \tau (\theta_f - \theta_i) = \Delta K = \frac{1}{2} I \omega_f^2 - \frac{1}{2} I \omega_i^2$$

• Power:

  $$P = \frac{dW}{dt} = \tau \omega$$

Conservation of Angular Momentum

Law of Conservation of Angular Momentum:
If the net external torque is zero ($\tau_{\text{net}} = 0$), then the total angular momentum of the system is conserved:

$$\vec{L} = \text{constant}$$