Physics Study Guide: Rotational Dynamics, Energy Conservation, Angular Momentum, Static Equilibrium, Circular Motion, and Gravity

Rotational Dynamics

Moment of Inertia

The moment of inertia quantifies an object's resistance to changes in rotational motion about a given axis. It depends on the mass distribution relative to the axis of rotation.

• Definition: $I = \sum m_i r_i^2$ (discrete masses), $I = \int r^2 dm$ (continuous mass distribution)

• Common Moments of Inertia:

  Object	Moment of Inertia	Axis
  Solid sphere	$I = \frac{2}{5}MR^2$	Through center of mass
  Solid cylinder	$I = \frac{1}{2}MR^2$	Through center of mass
  Rod	$I = \frac{1}{12}ML^2$	Through center of mass, perpendicular to length
  Rod	$I = \frac{1}{3}ML^2$	Through end, perpendicular to length

• Parallel Axis Theorem: $I = I_{cm} + Md^2$ (where $d$ is the distance from the center of mass axis)

Example: Calculating the moment of inertia for a rod about its end using $I = \frac{1}{3}ML^2$.

Rotational Kinematics and Dynamics

Rotational motion is described by angular displacement, velocity, and acceleration, analogous to linear motion.

• Angular velocity: $\omega = \frac{d\theta}{dt}$

• Angular acceleration: $\alpha = \frac{d\omega}{dt}$

• Rotational analog of Newton's Second Law: $\tau = I\alpha$

• Torque: $\tau = rF\sin\theta$

Example: Finding the angular acceleration of a rod with masses attached at its ends.

Energy Conservation

Conservation of Mechanical Energy

The law of conservation of energy states that the total mechanical energy (kinetic + potential) in a system remains constant if only conservative forces act.

• Mechanical energy: $E = K + U$

• Kinetic energy: $K = \frac{1}{2}mv^2$

• Potential energy (gravity): $U_g = mgh$

• Potential energy (spring): $U_s = \frac{1}{2}kx^2$

• Work by non-conservative forces: $W_{nc} = \Delta E$

Example: Calculating the speed of a mass after being pushed by a spring and frictional force.

Angular Momentum

Definition and Conservation

Angular momentum is a measure of the rotational motion of an object and is conserved in the absence of external torques.

• Definition: $\vec{L} = \vec{r} \times \vec{p}$

• For a particle: $L = mvr\sin\theta$

• For a rotating rigid body: $L = I\omega$

• Conservation: $L_{initial} = L_{final}$ if $\sum \tau_{ext} = 0$

Example: Calculating the angular momentum of a ball before and after collision with a rotating sphere.

Static Equilibrium

Conditions for Equilibrium

An object is in static equilibrium if the net force and net torque acting on it are zero.

• Translational equilibrium: $\sum \vec{F} = 0$

• Rotational equilibrium: $\sum \tau = 0$

• Application: Used to solve for unknown forces or masses in systems with pulleys, rods, and strings.

Example: Finding the mass of a rod suspended by a string at an angle using equilibrium conditions.

Circular Motion

Forces in Circular Motion

Objects moving in a circle experience a centripetal force directed toward the center of the circle.

• Centripetal force: $F_c = \frac{mv^2}{r}$

• Friction as centripetal force: $F_{friction} = \mu_s N$

• Maximum speed without sliding: $v_{max} = \sqrt{\mu_s g r}$

Example: Determining the shortest time for a car to make a turn without sliding.

Gravity and Circular Orbits

Universal Law of Gravitation

Newton's law of universal gravitation describes the attractive force between two masses.

• Gravitational force: $F = G\frac{m_1 m_2}{r^2}$

• Gravitational potential energy: $U = -G\frac{m_1 m_2}{r}$

• Orbital speed: $v = \sqrt{\frac{GM}{r}}$

• Period of orbit: $T = 2\pi \sqrt{\frac{r^3}{GM}}$

Example: Calculating the distance from the center of the moon to the center of the earth for a circular orbit, and the speed of a ball dropped onto the moon.

Summary Table: Key Equations

Topic	Equation
Moment of Inertia	$I = \sum m_i r_i^2$, $I = \int r^2 dm$
Rotational Kinematics	$\omega = \frac{d\theta}{dt}$, $\alpha = \frac{d\omega}{dt}$
Torque	$\tau = rF\sin\theta$
Energy Conservation	$E = K + U$, $K = \frac{1}{2}mv^2$, $U_g = mgh$, $U_s = \frac{1}{2}kx^2$
Angular Momentum	$L = I\omega$, $\vec{L} = \vec{r} \times \vec{p}$
Circular Motion	$F_c = \frac{mv^2}{r}$, $v_{max} = \sqrt{\mu_s g r}$
Gravity	$F = G\frac{m_1 m_2}{r^2}$, $U = -G\frac{m_1 m_2}{r}$

Additional info: These notes are based on a final exam equation sheet and a set of exam problems covering topics from rotational dynamics, energy conservation, angular momentum, static equilibrium, circular motion, and gravity/orbits. All equations are standard for introductory college physics and are suitable for exam preparation.