Potential Energy & Energy Conservation: Work, Kinetic Energy, and Systems

Potential Energy & Energy Conservation

Work and Kinetic Energy Recap

The concepts of work and kinetic energy are foundational in understanding energy transfer in physical systems. Work is done when a force causes displacement, and kinetic energy quantifies the energy of motion.

• Work (W): Defined as the product of the force magnitude, displacement, and the cosine of the angle between them. $W=|{F}||d{}|cos\phi$ Unit: Joule (J) = Newton·meter (N·m)

• Kinetic Energy (K): The energy associated with an object's motion. $K = \frac{1}{2}mv^2$ Unit: kg·m2/s2 = J

• Work-Kinetic Energy Theorem: The net work done on an object equals its change in kinetic energy. $W_{\text{total}} = \Delta K$

• Advantages: Useful for finding final speed after displacement. Disadvantages: Does not directly provide time information.

• Example: Calculating the speed of an object moving from point P to Q using work-energy principles.

Concept Check: Effects of Work on Kinetic Energy

When positive work is done on an object, its kinetic energy:

• Increases (since energy is transferred to the object).

• Negative work would decrease kinetic energy.

• If no work is done, kinetic energy remains the same.

Concept Check 2: Work on an Inclined Plane

When a crate is pulled up a rough inclined plane at constant speed:

• Friction does negative work (opposes motion).

• Net force work is zero (constant speed means no net acceleration).

• Normal force does zero work (perpendicular to displacement).

• Gravity does nonzero work (unless displacement is perpendicular to gravity).

Potential Energy and Systems

Definition of Potential Energy

Potential energy is defined for a system of multiple objects, based on their relative positions. It represents stored energy due to configuration.

• Gravitational Potential Energy (Ug): For a system of an object and Earth: $U_g = mgy$ Depends on: Mass (m) and height (y) above a reference point.

• Reference Height: The choice of y = 0 is arbitrary; only changes in height matter.

• Change in Potential Energy: $\Delta U_g = mg(y_f - y_i)$

• Relationship to Work: Work done by gravity is the negative of the change in potential energy: $\Delta U_g = -W_{\text{earth}}$

Solving Problems with Kinetic and Potential Energy

Energy conservation principles allow us to analyze systems involving both kinetic and potential energy.

• Generalized Work-Energy Principle: $K_i + U_{g,i} + W_{\text{other}} = K_f + U_{g,f}$ Where $W_{\text{other}}$ includes work by non-conservative forces (e.g., friction, tension).

• If only conservative forces act: $K_i + U_{g,i} = K_f + U_{g,f}$ Total mechanical energy is conserved.

• Example: Roller coaster system, where both kinetic and gravitational potential energy are considered.

Forces That Do Zero Work

Some forces do zero work because they act perpendicular to the direction of motion.

• Normal Force: On a body sliding along a surface, the normal force is perpendicular to displacement.

• Tension Force: On a pendulum bob, tension is perpendicular to the path of motion.

Summary Table: Work, Energy, and Forces

Quantity	Definition	Equation	Unit
Work (W)	Force times displacement	$W = |oldsymbol{F}| |oldsymbol{s}| \\cos\phi$	Joule (J)
Kinetic Energy (K)	Energy of motion	$K = \frac{1}{2}mv^2$	Joule (J)
Gravitational Potential Energy (Ug)	Energy due to height	$U_g = mgy$	Joule (J)
Work-Energy Theorem	Net work equals change in kinetic energy	$W_{\text{total}} = \Delta K$	Joule (J)
Energy Conservation	Total mechanical energy conserved (no non-conservative forces)	$K_i + U_{g,i} = K_f + U_{g,f}$	Joule (J)

Key Applications and Examples

• Roller Coaster: Analyzing energy changes as the coaster moves along the track, considering both kinetic and potential energy.

• Inclined Plane: Understanding work done by gravity, friction, and normal force.

• Pendulum: Tension force does zero work; energy changes are due to gravity.

Additional info: These notes cover the essential principles of work, kinetic energy, potential energy, and energy conservation, suitable for introductory college physics. The examples and equations provided are foundational for problem-solving in mechanics.