Rotation, Static Equilibrium, and Stress-Strain: PHYS 120 Lecture 19 Study Notes

Rotation and Angular Momentum

Angular Momentum in Rotational Motion

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

• Definition: Angular momentum (L) for a rotating object is given by $L = I \omega$, where I is the moment of inertia and $\omega$ is the angular velocity.

• Conservation of Angular Momentum: In the absence of external torques, the total angular momentum of a system remains constant: $L_1 = L_2$.

• Example: A 20.0-cm diameter, 1.0-kg solid disk rotating at 200 rpm is joined by a 1.0-kg circular loop. The final angular velocity $\omega_2$ is found using conservation of angular momentum:

  • Moment of inertia for solid disk: $I_1 = \frac{1}{2}MR^2$

  • Moment of inertia for disk + loop: $I_2 = \frac{1}{2}MR^2 + MR^2 = \frac{3}{2}MR^2$

  • Conservation: $I_1 \omega_1 = I_2 \omega_2$

  • Solving: $\omega_2 = \frac{1}{3}\omega_1$

Additional info: The example demonstrates how angular velocity decreases when the moment of inertia increases, keeping angular momentum constant.

Static Equilibrium

Conditions for Static Equilibrium

Static equilibrium occurs when an object is at rest and all the forces and torques acting on it are balanced. This is essential for analyzing structures and systems that do not move.

• Definition: An object is in static equilibrium if it is not moving (translating or rotating) and all forces and torques are balanced.

• Conditions:

  • No translation: $\sum F_x = 0$, $\sum F_y = 0$

  • No rotation: $\sum \tau = 0$ (sum of torques about any pivot)

• Free-Body Diagrams (FBDs): Drawing FBDs helps identify all forces and torques acting on the system.

• Example: A head held erect requires neck muscles to exert a force to maintain equilibrium. Calculations involve balancing torques and forces:

  • Torque by muscles: $\tau_M = F_M r_M$

  • Torque by gravity: $\tau_g = mg r_g$

  • Equilibrium: $\sum \tau = -\tau_g + \tau_M = 0$

  • Force at pivot: $F_J = F_M + mg$

Problem-Solving Strategy for Equilibrium

Systematic Approach

Analyzing equilibrium problems requires a step-by-step approach to identify forces, torques, and the type of motion.

• Step 1: Identify all objects in the system and draw diagrams (FBDs).

• Step 2: Determine if the system is translating, rotating, both, or neither.

• Step 3: Apply the second law for rotation ($\sum \tau = 0$) and translation ($\sum F = 0$) as appropriate.

• Step 4: Identify the relevant radii and axes for torque calculations.

Type of Motion	Relevant Law
Only Translating	Newton's Second Law ($\sum F = ma$)
Only Rotating	Rotational Dynamics ($\sum \tau = I\alpha$)
Both	Combine Translational & Rotational Equilibrium
Neither	Static Equilibrium ($\sum F = 0$, $\sum \tau = 0$)

Applications of Static Equilibrium

Leaning Ladder Problem

Static equilibrium principles are used to analyze objects like ladders leaning against walls, where friction and forces must be considered.

• Example: A 2-m, 20-kg ladder leans against a frictionless wall. The floor has a coefficient of friction $\mu$. The ladder can have a maximum angle $\theta = 60^\circ$ before sliding. Find $\mu$.

• Equations:

  • $\sum F_x = 0$: $F_{fr} = F_{N2}$

  • $\sum F_y = 0$: $F_{N1} = mg$

  • $\sum \tau = 0$: $\tau_{N2} + \tau_{fr} = (mg)\frac{l}{2}\sin\theta$

  • Solving for $\mu$: $\mu = \frac{1}{2} \frac{\cos\theta}{\sin\theta} = \frac{1}{2 \tan\theta}$

  • For $\theta = 60^\circ$, $\mu = 0.28$

Types of Equilibrium

Stable, Unstable, and Neutral Equilibrium

Objects can be in different types of equilibrium depending on their response to small disturbances.

• Stable Equilibrium: The object returns to its original position after a small perturbation.

• Unstable Equilibrium: The object moves further away from its original position after a disturbance.

• Neutral Equilibrium: The object stays in its new position after being disturbed.

• Example: A ball at the bottom of a valley (stable), at the top of a hill (unstable), or on a flat surface (neutral).

Stress, Strain, and Elasticity

Mechanical Properties of Materials

Stress and strain describe how materials deform under applied forces. Elasticity quantifies a material's ability to return to its original shape.

• Stress: Force applied per unit area: $\text{Stress} = \frac{F}{A}$

• Strain: Relative change in length: $\text{Strain} = \frac{\Delta l}{l_0}$

• Young's Modulus (E): A constant that relates stress and strain for a material: $E = \frac{\text{Stress}}{\text{Strain}}$

• Types of Stress:

  • Tensile (Tension): Pulling forces

  • Compression: Pushing forces

  • Bulk Stress: Forces applied in all directions

• Example: Calculating the change in length of a pencil lead (graphite) with $E = 1 \times 10^9 \text{ N/m}^2$ when a force of 4 N is applied:

  • Diameter $d = 0.5$ mm, $r = 0.00025$ m

  • Length $l_0 = 0.06$ m

  • Area $A = \pi r^2$

  • Change in length: $\Delta l = \frac{F}{E \pi r^2} l_0$

  • Calculation yields $\Delta l = 1.22 \times 10^{-3}$ m

Additional info: Young's modulus is a measure of stiffness; higher values indicate less deformation under the same force.

Summary Table: Types of Equilibrium

Type	Description	Response to Disturbance
Stable	Returns to original position	Restores equilibrium
Unstable	Moves away from original position	Equilibrium lost
Neutral	Stays in new position	No change