week 9 handout

Rotational Motion

Conservation of Angular Momentum

Angular momentum is a fundamental concept in rotational dynamics, describing the rotational analog of linear momentum. For an isolated system, angular momentum is conserved unless acted upon by an external torque.

• Angular Momentum (L): Defined as $L = I \omega$, where I is the moment of inertia and \omega is the angular velocity.

• Conservation Principle: If no external torque acts on a system, its total angular momentum remains constant: $L_i = L_f$.

• Example: An ice skater spinning with arms extended pulls her arms in, reducing her moment of inertia and increasing her angular velocity to conserve angular momentum.

Calculation Example:

• Initial moment of inertia: $I_i = 4.0\ \mathrm{kg\ m^2}$

• Initial angular velocity: $\omega_i = 2.0\ \mathrm{rad/s}$

• Final moment of inertia: $I_f = 1.0\ \mathrm{kg\ m^2}$

• Final angular velocity: $\omega_f = \frac{I_i \omega_i}{I_f} = \frac{4.0 \times 2.0}{1.0} = 8.0\ \mathrm{rad/s}$

Rotational Kinetic Energy

Rotational kinetic energy is the energy due to the rotation of an object and depends on its moment of inertia and angular velocity.

• Formula: $KE_{rot} = \frac{1}{2} I \omega^2$

• Example Calculation:

  • Initial: $KE_{rot,i} = \frac{1}{2} (4.0) (2.0)^2 = 8\ \mathrm{J}$

  • Final: $KE_{rot,f} = \frac{1}{2} (1.0) (8.0)^2 = 32\ \mathrm{J}$

• Application: When the skater pulls in her arms, her rotational kinetic energy increases, even though angular momentum is conserved. This is because work is done to pull in the arms.

Static Equilibrium

Definition and Conditions

Static equilibrium occurs when an object is at rest and the sum of all forces and torques acting on it is zero. This ensures the object does not translate or rotate.

• Translational Equilibrium: $\sum \vec{F} = 0$ (sum of all forces is zero)

• Rotational Equilibrium: $\sum \tau = 0$ (sum of all torques is zero)

• Static vs. Dynamic Equilibrium:

  • Static Equilibrium: Object is at rest ($\vec{v} = 0$)

  • Dynamic Equilibrium: Object moves with constant velocity ($\vec{a} = 0$, $\vec{v} \neq 0$)

Examples of Equilibrium Situations

• Static Equilibrium: A barbell held overhead, a steel beam at rest, a parked car.

• Dynamic Equilibrium: A steel beam lifted at constant speed, a jet flying at constant altitude.

• Not in Equilibrium: A car that has just increased speed, a jet climbing or descending.

Force and Torque Analysis

To determine equilibrium, analyze the forces and torques acting on the object. For rotational equilibrium, consider the sum of torques about any axis.

• Force Equations: $\sum F_x = 0$, $\sum F_y = 0$

• Torque Equation: $\sum \tau = 0$

• Example: For a disk acted on by three forces at different points, set up equations for both force and torque to check equilibrium.

Table: Types of Equilibrium

Situation	Type of Equilibrium	Velocity	Acceleration
Barbell held overhead	Static	0	0
Steel beam lifted at constant speed	Dynamic	Constant	0
Car parked	Static	0	0
Jet flying at constant altitude	Dynamic	Constant	0
Car accelerating	Not in equilibrium	Increasing	Nonzero

Applications

• Engineering: Design of stable structures, bridges, and buildings.

• Physics: Analysis of forces in statics problems, rotational dynamics.

• Everyday Life: Balancing objects, understanding why things tip over.

Additional info: The notes include handwritten solutions and explanations for rotational motion and static equilibrium, suitable for introductory college physics. All equations and examples have been expanded for clarity and completeness.