Work, Energy, and Momentum: Exam 2 Study Guide

Chapter 6: Work and the Work-Energy Theorem

Identifying Positive, Negative, and Zero Work

Work is a measure of energy transfer that occurs when a force acts upon an object to cause displacement. The sign of work (positive, negative, or zero) depends on the direction of the force relative to the displacement.

• Positive Work: Force and displacement are in the same direction. Example: Lifting a box upward.

• Negative Work: Force and displacement are in opposite directions. Example: Lowering a box gently to the ground.

• Zero Work: Force is perpendicular to displacement, or there is no displacement. Example: Carrying a bag at constant height horizontally.

Formula:

$W = Fd \cos\theta$

where $W$ is work, $F$ is the magnitude of the force, $d$ is the displacement, and $\theta$ is the angle between the force and displacement vectors.

Calculating Work Done

To calculate work, multiply the component of force in the direction of displacement by the magnitude of the displacement.

• For constant force: $W = Fd \cos\theta$

• For variable force: $W = \int_{x_i}^{x_f} F(x) dx$

Example: Pushing a crate across a floor with a constant force at an angle.

Work-Energy Theorem

The work-energy theorem relates the net work done on an object to its change in kinetic energy.

• Statement: The net work done by all forces on an object equals the change in its kinetic energy.

$W_{net} = \Delta K = K_f - K_i = \frac{1}{2}mv_f^2 - \frac{1}{2}mv_i^2$

Application: Used to solve problems involving forces and motion without directly using Newton's laws.

Work Done by a Varying Force

When the force varies with position, work is found by integrating the force over the path of motion.

• Formula: $W = \int_{x_i}^{x_f} F(x) dx$

• Graphical Interpretation: The area under the force vs. displacement curve represents the work done.

Example: Stretching a spring, where force increases linearly with displacement.

Chapter 7: Conservation of Energy

Conservation of Energy with Only Gravity

When only gravity does work, mechanical energy (kinetic + potential) is conserved.

• Mechanical Energy: $E = K + U_g$

• Gravitational Potential Energy: $U_g = mgh$

• Conservation Law: $K_i + U_{g,i} = K_f + U_{g,f}$

Example: A ball thrown upward; as it rises, kinetic energy converts to potential energy.

Conservation of Energy with Gravity and Friction

When friction is present, mechanical energy is not conserved, but the work done by friction must be included.

• Modified Conservation Law: $K_i + U_{i} + W_{nc} = K_f + U_{f}$

• $W_{nc}$ is the work done by non-conservative forces (e.g., friction).

Example: A block sliding down a ramp with friction loses mechanical energy to heat.

Elastic Potential Energy

Elastic potential energy is stored in objects like springs when they are compressed or stretched.

• Formula: $U_{el} = \frac{1}{2}kx^2$

• $k$ is the spring constant, $x$ is the displacement from equilibrium.

Example: Compressing a spring in a toy gun stores energy that is released when fired.

Energy Conservation with Conservative Forces

Conservative forces (like gravity and elastic forces) allow for the conservation of mechanical energy.

• Conservative Force: A force for which the work done is path-independent and can be fully recovered.

• Examples: Gravitational force, spring (elastic) force.

• Conservation Law: $E_{mech, i} = E_{mech, f}$ (if only conservative forces act)

Example: A mass oscillating on a spring exchanges kinetic and potential energy, but total mechanical energy remains constant.

Chapter 8: Momentum and Collisions

Force-Momentum-Impulse Relationships

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

• 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 bat hitting a baseball applies an impulse, changing the ball's momentum.

Conservation of Momentum

In a closed system with no external forces, the total momentum before an event equals the total momentum after.

• Law of Conservation of Momentum: $\sum \vec{p}_{i} = \sum \vec{p}_{f}$

• Applies to collisions and explosions.

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

Elastic Collisions in One and Two Dimensions

In elastic collisions, both momentum and kinetic energy are conserved.

• One Dimension: $m_1v_{1i} + m_2v_{2i} = m_1v_{1f} + m_2v_{2f}$

• Kinetic Energy Conservation: $\frac{1}{2}m_1v_{1i}^2 + \frac{1}{2}m_2v_{2i}^2 = \frac{1}{2}m_1v_{1f}^2 + \frac{1}{2}m_2v_{2f}^2$

• Two Dimensions: Apply conservation laws separately to x and y components.

Example: Billiard balls colliding on a pool table.

Inelastic Collisions in One and Two Dimensions

In inelastic collisions, momentum is conserved but kinetic energy is not. In a perfectly inelastic collision, objects stick together after the collision.

• Momentum Conservation: $m_1v_{1i} + m_2v_{2i} = (m_1 + m_2)v_f$ (for perfectly inelastic)

• Kinetic energy is lost to deformation, heat, or sound.

Example: Two cars colliding and sticking together after impact.

Summary Table: Types of Collisions

Type of Collision	Momentum Conserved?	Kinetic Energy Conserved?	Example
Elastic	Yes	Yes	Billiard balls
Inelastic	Yes	No	Car crash (cars stick together)
Perfectly Inelastic	Yes	No	Clay balls sticking together