Work, Energy, and Momentum: Study Guide for PHYS 141 Exam 2 (Chapters 6–8)

Work, Energy, and Momentum

Chapter 6: Work and the Work-Energy Theorem

This chapter introduces the concepts of work, kinetic energy, and the work-energy theorem, which are foundational for understanding how forces affect the motion and energy of objects. The concept of power, which measures the rate at which work is done, is also discussed.

• Work: Work is done when a force causes a displacement of an object. For a constant force F acting over a displacement d at an angle θ to the displacement:

$W = F d \cos \theta$

• Kinetic Energy (KE): The energy associated with the motion of an object. For an object of mass m and speed v:

$K = \frac{1}{2} m v^2$

• The Work-Energy Theorem: The net work done on an object is equal to its change in kinetic energy:

$W_{\text{net}} = \Delta K = K_f - K_i$

• Work by a Varying Force: When the force varies with position, work is calculated as the area under the force vs. displacement curve, or by integration:

$W = \int_{x_i}^{x_f} F(x) \, dx$

• Power: Power is the rate at which work is done.

  • In terms of work and time:

$P = \frac{W}{\Delta t}$

• In terms of force and velocity (for constant force):

$P = \vec{F} \cdot \vec{v}$

• Example: A 10 kg box is pushed 5 m across a floor by a 20 N force at 0° to the displacement. The work done is $W = 20 \times 5 \times \cos 0° = 100$ J.

Chapter 7: Potential Energy and Conservation of Energy

This chapter explores potential energy, the principle of energy conservation, and the distinction between conservative and nonconservative forces. It also covers how energy is stored and transformed in physical systems.

• Potential Energy (U): Energy stored due to an object's position or configuration.

• Gravitational Potential Energy: For an object of mass m at height y above a reference point:

$U_g = m g y$

• Elastic Potential Energy: For a spring with force constant k and displacement x from equilibrium:

$U_s = \frac{1}{2} k x^2$

• Conservation of Mechanical Energy: In the absence of nonconservative forces (like friction), the total mechanical energy (kinetic + potential) is conserved:

$K_i + U_i = K_f + U_f$

• Nonconservative Forces: If nonconservative forces (e.g., friction) do work, mechanical energy is not conserved, but total energy is:

$K_i + U_i + W_{\text{nc}} = K_f + U_f$

• Work and Potential Energy (Conservative Forces): The work done by a conservative force equals the negative change in potential energy:

$W_{\text{cons}} = -\Delta U$

• Force and Potential Energy: The force associated with a potential energy function is the negative gradient of the potential energy:

$F_x = -\frac{dU}{dx}$

• Energy Graphs: Graphs of potential and kinetic energy can be used to analyze motion and equilibrium points.

• Example: A 2 kg mass is lifted 3 m. The change in gravitational potential energy is $\Delta U_g = 2 \times 9.8 \times 3 = 58.8$ J.

Chapter 8: Momentum, Impulse, and Collisions

This chapter introduces momentum and impulse, and applies these concepts to analyze collisions and the motion of systems of particles. The center of mass and its significance are also discussed.

• Momentum (p): The product of an object's mass and velocity:

$\vec{p} = m \vec{v}$

• Impulse (J): The product of the average force and the time interval over which it acts; equals the change in momentum:

$\vec{J} = \vec{F}_{\text{avg}} \Delta t = \Delta \vec{p}$

• Impulse-Momentum Theorem: The net impulse on an object equals its change in momentum.

• Conservation of Momentum: In a closed system with no external forces, total momentum is conserved:

$\sum \vec{p}_{\text{initial}} = \sum \vec{p}_{\text{final}}$

• Collisions:

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

  • Inelastic Collisions: Momentum is conserved, but kinetic energy is not. In a perfectly inelastic collision, objects stick together after the collision.

• Center of Mass (CM): The weighted average position of all the particles in a system:

$x_{\text{CM}} = \frac{\sum m_i x_i}{\sum m_i}$

• Newton's Second Law (Momentum Form): The net external force equals the rate of change of momentum:

$\vec{F}_{\text{net}} = \frac{d\vec{p}}{dt}$

• Example: Two ice skaters push off from each other. If one has twice the mass of the other, her velocity after the push will be half as large in magnitude but in the opposite direction, conserving total momentum.

Summary Table: Key Quantities and Equations

Quantity	Symbol	Equation	Units (SI)
Work	W	$W = F d \cos \theta$	Joule (J)
Kinetic Energy	K	$K = \frac{1}{2} m v^2$	Joule (J)
Potential Energy (Gravitational)	$U_g$	$U_g = m g y$	Joule (J)
Potential Energy (Elastic)	$U_s$	$U_s = \frac{1}{2} k x^2$	Joule (J)
Momentum	p	$\vec{p} = m \vec{v}$	kg·m/s
Impulse	J	$\vec{J} = \vec{F}_{\text{avg}} \Delta t$	kg·m/s
Power	P	$P = \frac{W}{\Delta t}$, $P = \vec{F} \cdot \vec{v}$	Watt (W)

Additional info: This guide summarizes the main concepts and equations from Chapters 6–8, as outlined in the PHYS 141 Exam 2 Study Guide. Students should also review prior material on 1D motion, Newton's laws, and acceleration in 2D, as these may appear on the exam.