Exam 4 Review: Equilibrium, Elasticity, Momentum, and Energy

Equilibrium and Elasticity

Torque and Static Equilibrium

Understanding the conditions for static equilibrium is essential for analyzing objects at rest. These conditions ensure that an object does not translate or rotate.

• Static Equilibrium: An object is in static equilibrium if the net force and net torque about any axis are both zero.

• Equations:

  • Sum of forces: $\sum \vec{F} = 0$

  • Sum of torques: $\sum \tau = 0$

• Example: A ladder leaning against a wall, with forces acting at the base and top, can be analyzed using these equilibrium conditions.

Stability and Center of Gravity

Stability depends on the location of the center of gravity relative to the base of support.

• Stable Equilibrium: If the center of gravity is above the base of support, the object returns to its original position after a small displacement.

• Unstable Equilibrium: If displaced, the object moves further from its original position.

• Critical Angle: The maximum angle before an object tips over.

• Example: Balancing a pencil on its tip demonstrates unstable equilibrium.

Elasticity

Elasticity describes how materials deform and return to their original shape when forces are applied and removed.

• Young's Modulus: $Y = \frac{\text{Stress}}{\text{Strain}}$

• Shear Modulus: $S = \frac{\text{Shear Stress}}{\text{Shear Strain}}$

• Bulk Modulus: $B = -\frac{\Delta P}{\frac{\Delta V}{V}}$

• Example: Stretching a steel wire and measuring its elongation.

Momentum

Impulse and Momentum

Momentum is a measure of an object's motion, and impulse is the change in momentum resulting from a force applied over time.

• Linear Momentum: $\vec{p} = m\vec{v}$

• Impulse: $\vec{J} = \Delta \vec{p} = \int \vec{F} dt$

• Impulse-Momentum Theorem: $\vec{J} = \Delta \vec{p}$

• Example: A baseball bat striking a ball delivers an impulse, changing the ball's momentum.

Conservation of Momentum

In the absence of external forces, the total momentum of a system remains constant.

• Equation: $m_1 v_{1i} + m_2 v_{2i} = m_1 v_{1f} + m_2 v_{2f}$

• Example: Two ice skaters push off from each other and move in opposite directions.

Collisions

Collisions are classified as elastic or inelastic based on whether kinetic energy is conserved.

• Elastic Collision: Both momentum and kinetic energy are conserved.

• Inelastic Collision: Momentum is conserved, but kinetic energy is not. Perfectly inelastic collisions involve objects sticking together.

• Example: Two carts colliding on a track and sticking together (perfectly inelastic).

Momentum in Two Dimensions

Momentum conservation applies independently in each direction.

• Equation: $\sum \vec{p}_{\text{initial}} = \sum \vec{p}_{\text{final}}$ (vector sum)

• Example: Billiard balls colliding at an angle.

Angular Momentum

Angular momentum is the rotational analog of linear momentum and is conserved in the absence of external torques.

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

• Conservation: $\sum \vec{L}_{\text{initial}} = \sum \vec{L}_{\text{final}}$

• Example: A figure skater spinning faster by pulling in her arms.

Energy and Work

Work and Kinetic Energy

Work is done when a force causes displacement. The work-energy theorem relates work to changes in kinetic energy.

• Work: $W = \vec{F} \cdot \vec{d}$

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

• Work-Energy Theorem: $W_{\text{net}} = \Delta K$

• Example: Lifting a box vertically increases its gravitational potential energy.

Potential Energy and Conservation of Energy

Potential energy is stored energy due to position or configuration. The law of conservation of energy states that the total energy in an isolated system remains constant.

• Gravitational Potential Energy: $U_g = mgh$

• Elastic Potential Energy: $U_s = \frac{1}{2}kx^2$

• Conservation of Mechanical Energy: $K_i + U_i = K_f + U_f$

• Example: A pendulum swinging back and forth exchanges kinetic and potential energy.

Power

Power is the rate at which work is done or energy is transferred.

• Equation: $P = \frac{W}{t}$

• Instantaneous Power: $P = \vec{F} \cdot \vec{v}$

• Example: An elevator lifting passengers at a constant speed.

Energy Transformation and Dissipation

Energy can be transformed from one form to another, and some energy is often dissipated as heat due to friction or other non-conservative forces.

• Example: A car braking to a stop converts kinetic energy into thermal energy in the brakes.

Summary Table: Key Quantities and Equations

Quantity	Symbol	Equation	SI Unit
Linear Momentum	\( \vec{p} \)	$\vec{p} = m\vec{v}$	kg·m/s
Impulse	\( \vec{J} \)	$\vec{J} = \int \vec{F} dt$	N·s
Work	W	$W = \vec{F} \cdot \vec{d}$	J (joule)
Kinetic Energy	K	$K = \frac{1}{2}mv^2$	J
Potential Energy (gravitational)	$U_g$	$U_g = mgh$	J
Elastic Potential Energy	$U_s$	$U_s = \frac{1}{2}kx^2$	J
Power	P	$P = \frac{W}{t}$	W (watt)
Angular Momentum	\( \vec{L} \)	$\vec{L} = \vec{r} \times \vec{p}$	kg·m2/s

Additional info: Some explanations and examples were expanded for clarity and completeness based on standard physics curriculum.