AP Physics 1 Comprehensive Notes - Unit 6: Energy and Momentum of Rotating Systems

AP Physics 1 Comprehensive Notes - Unit 6: Energy and Momentum of Rotating Systems

Unit 6: Energy and Momentum of Rotating Systems

Moment of Inertia or Rotational Mass

For a system of particles: $I = \sum_{i} m_i r_i^2$
Dimensions: $I = \sum_{i} m_i r_i^2 \Rightarrow kg \cdot m^2$
For a rigid object, the moment of inertia depends on shape:
- Solid Cylinder: $I_{\text{solid cylinder}} = \frac{1}{2} MR^2$
- Thin Hoop: $I_{\text{thin hoop}} = MR^2$
- Solid Sphere: $I_{\text{solid sphere}} = \frac{2}{5} MR^2$
- Thin Spherical Shell: $I_{\text{thin spherical shell}} = \frac{2}{3} MR^2$
- Rod (center of mass): $I_{\text{rod}} = \frac{1}{12} ML^2$
- Rod (about end): $I_{\text{rod about end}} = \frac{1}{3} ML^2$

Rotational Kinetic Energy

$KE_{\text{rot}} = \frac{1}{2} I \omega^2$
Like translational energy, units: Joules (J).
Rolling without slipping: Objects rolling downhill have both translational and rotational kinetic energy. $v_{\text{cm}} = R\omega$

Angular Momentum

$\mathbf{L} = I \omega$
Units: $\mathbf{L} = I \omega \Rightarrow (kg \cdot m^2) \left(\frac{\text{rad}}{s}\right) = kg \cdot m^2/s$

Angular Impulse

$\Delta \mathbf{L} = \mathbf{\tau}_{\text{impact}} \Delta t = \text{Angular Impulse}$
Units: $\Delta \mathbf{L} = \mathbf{\tau}_{\text{impact}} \Delta t \Rightarrow N \cdot m \cdot s$

Comparison of Linear and Rotational Motion

Name	Linear	Rotational
Displacement	$\Delta x = x_f - x_i$	$\Delta \theta = \theta_f - \theta_i$
Velocity	$\bar{v}_{\text{avg}} = \frac{\Delta x}{\Delta t}$	$\bar{\omega}_{\text{avg}} = \frac{\Delta \theta}{\Delta t}$
Acceleration	$\bar{a}_{\text{avg}} = \frac{\Delta v}{\Delta t}$	$\bar{\alpha}_{\text{avg}} = \frac{\Delta \omega}{\Delta t}$
Uniformly Accelerated Motion (UAM)	$v_f = v_i + at$	$\omega_f = \omega_i + \alpha t$
	$x_f = x_i + v_i t + \frac{1}{2} at^2$	$\theta_f = \theta_i + \omega_i t + \frac{1}{2} \alpha t^2$
	$v_f^2 = v_i^2 + 2a(x_f - x_i)$	$\omega_f^2 = \omega_i^2 + 2\alpha(\theta_f - \theta_i)$
Mass	Mass	$I_{\text{particles}} = \sum_i m_i r_i^2$
Kinetic Energy	$KE_{\text{translational}} = \frac{1}{2} m v^2$	$KE_{\text{rotational}} = \frac{1}{2} I \omega^2$
Newton’s Second Law	$\sum F = ma$	$\sum \tau = I \alpha$
Force / Torque	Force	$\tau = r \times F$
Power	$P_{\text{translational}} = \mathbf{F} \cdot \mathbf{v}$	$P_{\text{rotational}} = \mathbf{\tau} \cdot \mathbf{\omega}$
Momentum	$\mathbf{p} = m\mathbf{v}$	$\mathbf{L}_{\text{particle}} = r \times \mathbf{p}$ , $\mathbf{L}_{\text{object}} = I\omega$