$W = Fd\cos\theta$

• Force must have component parallel to displacement
• Positive work: force has component in direction of motion
• Negative work: force has component opposite to motion
• Zero work: force perpendicular to displacement
• Scalar quantity (no direction)
• Units: Joule (J)

• $W = \int F_x\,dx$ (variable force)
• Graphically: Area under F vs. position curve
• Spring force: $W = \int_0^x kx'\,dx' = \frac{1}{2}kx^2$
• Negative work when spring compresses

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

• Net work equals change in kinetic energy
• $K = \frac{1}{2}mv^2$
• Work-energy theorem: $W_{net} = \Delta K$
• Applies even when forces vary

• Conservative forces:
• Work is path-independent
• Work around closed path = 0
• Can define potential energy
• Examples: gravity, spring force, electric force
• Non-conservative forces:
• Work depends on path taken
• Friction, air resistance
• Cannot define potential energy
• Energy dissipated as thermal energy

$U_g = mgh$

• Depends on height above reference level
• Reference level chosen arbitrarily
• Negative work by gravity when object rises
• Positive work by gravity when object falls
• Near Earth's surface (g constant)

$U_s = \frac{1}{2}kx^2$

• Energy stored in compressed/stretched spring
• Zero at equilibrium position
• Positive when compressed or stretched
• Proportional to square of displacement
• Independent of gravitational effects

• $E_{mechanical} = K + U = \text{constant (no friction)}$
• $K_i + U_i = K_f + U_f$
• Holds when only conservative forces do work
• Applies to isolated systems (no external work)

• Visual representation of energy transformations
• Bar heights represent amounts of different energy types
• Shows energy conservation: total height constant
• Friction converts mechanical to thermal energy
• Useful for tracking energy through multi-stage processes

• $W_{nc} = \Delta E_{mechanical} = \Delta K + \Delta U$
• Friction does negative work (converts mechanical to thermal)
• Air resistance does negative work
• Total energy conserved (mechanical + thermal)

$K_i + U_i + W_{nc} = K_f + U_f + \Delta E_{thermal}$

• Rate of doing work or transferring energy
• $P = \frac{W}{t}$ (average power)
• Units: Watt (W) = J/s

Instantaneous Power Formula P=Fv: $P = Fv\cos\theta$

• $P = F\cdot v$ (force parallel to velocity)
• Instantaneous power at specific instant
• Engine power: $P = \vec{F} \cdot \vec{v}$
• Important for motors and engines

$P = Fv\cos\theta$

• $P = F\cdot v$ (force parallel to velocity)
• Instantaneous power at specific instant
• Engine power: $P = \vec{F} \cdot \vec{v}$
• Important for motors and engines

Conservative vs Non-Conservative Forces:

• Conservative forces:
• Work is path-independent
• Work around closed path = 0
• Can define potential energy
• Examples: gravity, spring force, electric force
• Non-conservative forces:
• Work depends on path taken
• Friction, air resistance
• Cannot define potential energy
• Energy dissipated as thermal energy

Gravitational Potential Energy: $U_g = mgh$

• Depends on height above reference level
• Reference level chosen arbitrarily
• Negative work by gravity when object rises
• Positive work by gravity when object falls
• Near Earth's surface (g constant)

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

• Energy stored in compressed/stretched spring
• Zero at equilibrium position
• Positive when compressed or stretched
• Proportional to square of displacement
• Independent of gravitational effects