Work, Kinetic Energy, and Conservation of Energy: Study Notes

Work and Kinetic Energy

Introduction to Energy

Energy is a fundamental concept in physics, representing the ability to do work. The universe is composed of two elemental building blocks: mass and energy. These are related by the famous equation:

• Mass-Energy Conversion: $E = mc^2$

Energy cannot be created nor destroyed; it can only be transferred or converted between different forms (the principle of conservation of energy).

Types of Energy

• Kinetic Energy: Energy related to motion. Examples include:

  • Motion of macroscopic objects

  • Heat

  • Light

  • Electric current

• Potential Energy: Energy related to position and force. Examples include:

  • Gravitational

  • Electrical

  • Magnetic

  • Nuclear

Kinetic energy requires a nonzero velocity, while potential energy requires a force acting on the object.

Work

Definition of Work

Work (W) is the process of transferring energy to or from a system by mechanical means. Doing work is equivalent to exchanging energy.

• Positive Work: Energy is added to the system.

• Negative Work: Energy is removed from the system.

Mathematically, work is defined as:

• $oxed{W = F riangle x ext{ (if force and displacement are parallel)}}$

• $oxed{W = F riangle x ext{cos} heta}$ (if force is at an angle $heta$ to displacement)

Unit of work: Joule (J)

Work and Displacement

• If there is no displacement ($riangle x = 0$), no work is done, regardless of the force applied.

• Example: Pushing against a wall with 200 N force but no movement results in zero work.

Dot Product and Work

Work is a scalar quantity obtained from the dot product of two vectors: force and displacement.

• $W = extbf{F} ullet riangle extbf{x} = | extbf{F}| | riangle extbf{x}| ext{cos} heta$

Sign of Work

• If $heta = 0^ ext{o}$ (force and displacement in same direction): $W > 0$

• If $0^ ext{o} < heta < 90^ ext{o}$: $W > 0$

• If $heta = 90^ ext{o}$: $W = 0$

• If $90^ ext{o} < heta ext{ or } heta = 180^ ext{o}$ (force opposite to displacement): $W < 0$

Kinetic Energy

Definition and Formula

Kinetic energy (KE) is the energy of motion. All moving objects possess kinetic energy.

• $oxed{KE = rac{1}{2}mv^2}$

Change in kinetic energy ($riangle KE$) is given by:

• $riangle KE = KE_f - KE_i = rac{1}{2}m(v_f^2 - v_i^2)$

Work-Kinetic Energy Theorem

The work done by the net force on an object is equal to the change in its kinetic energy:

• $oxed{W_{net} = riangle KE}$

Cases:

• Speeding up: $v_f > v_i$ → $W_{net} > 0$ (increase in kinetic energy)

• Slowing down: $v_f < v_i$ → $W_{net} < 0$ (decrease in kinetic energy)

• Constant speed: $v_f = v_i$ → $W_{net} = 0$ (no change in kinetic energy)

Potential Energy

Gravitational Potential Energy (GPE)

Potential energy due to an object's position in a gravitational field:

• $oxed{GPE = mgh}$

• Where $m$ is mass, $g$ is acceleration due to gravity, and $h$ is height above a reference point.

Conservative and Nonconservative Forces

Conservative Forces

A conservative force is one for which the work done is independent of the path taken. The work done by conservative forces equals the decrease in the corresponding potential energy.

• Example: Gravitational force

• $W_c = - riangle PE$

Nonconservative Forces

A nonconservative force is one for which the work done depends on the path. Examples include friction, tension, and air resistance.

• Work done by nonconservative forces represents energy exchanged with objects outside the system.

• Potential energy cannot be defined for nonconservative forces.

Work-Energy Theorem (General Case)

When both conservative and nonconservative forces act:

• $W_{net} = riangle K = W_c + W_{nc}$

Total Mechanical Energy and Conservation

Total Mechanical Energy

The total mechanical energy of an object is the sum of its potential energy ($U$) and kinetic energy ($K$):

• $oxed{E = PE + KE}$

• If only gravity acts: $E = mgh + rac{1}{2}mv^2$

Conservation of Mechanical Energy

If only conservative forces do work, the total mechanical energy remains constant:

• $riangle E = riangle PE + riangle KE = 0$

If nonconservative forces do work:

• $riangle E = W_{nc}$

Worked Examples

Example 1: Calculating Work

• Force $F_0 = 10$ N, displacement $riangle x = 2$ m, angle $heta = 60^ ext{o}$

• Case 1 (force parallel): $W = F_0 riangle x = 10 imes 2 = 20$ J

• Case 2 (force at angle): $W = F_0 riangle x ext{cos} 60^ ext{o} = 10 imes 2 imes 0.5 = 10$ J

Example 2: Final Speed of a Box

• Net force $F_{net} = 20$ N $-$ 10 N $= 10$ N

• Displacement $riangle x = 4$ m, mass $m = 5$ kg

• $W = F_{net} riangle x = 10 imes 4 = 40$ J

• $W = riangle KE = rac{1}{2} m v^2$

• $40 = rac{1}{2} imes 5 imes v^2 ightarrow v = 4$ m/s

Example 3: Conservation of Energy on a Ramp

• Parcel mass $m = 5$ kg, height $h = 2$ m

• $PE = mgh = 5 imes 9.8 imes 2 = 98$ J

• If no friction, $PE = KE$ at bottom: $KE = 98$ J

• $v = ext{sqrt}(2gh) = ext{sqrt}(2 imes 9.8 imes 2) = 6.26$ m/s$

Summary Table: Conservative vs Nonconservative Forces

Type of Force	Path Dependence	Potential Energy Defined?	Examples
Conservative	No	Yes	Gravity, Spring
Nonconservative	Yes	No	Friction, Air Resistance, Tension

Key Equations

• $E = mc^2$

• $KE = rac{1}{2}mv^2$

• $GPE = mgh$

• $W = F riangle x ext{cos} heta$

• $W_{net} = riangle KE$

• $E_{total} = PE + KE$

• $riangle E = W_{nc}$

Conceptual Questions

• Work is only done when a force causes displacement.

• Conservative forces do not change total mechanical energy; nonconservative forces do.

• All objects falling the same vertical distance (without resistance) have the same net work done by all forces.

Additional info: These notes expand on the original slides by providing full definitions, formulas, and context for each concept, ensuring completeness and clarity for exam preparation.