Energy of motion.

$K = \frac{1}{2}mv^2$

Units: Joules (J)

Properties:

• Always positive (or zero)
• Increases with mass and speed
• Frame-dependent (different in different reference frames)

Work is energy transfer by force through displacement.

$W = \vec{F} \cdot \vec{d} = Fd\cos\theta$

Where:

• F = force magnitude
• d = displacement magnitude
• $\theta$ = angle between force and displacement

Units: Joules (J) = Newton - meter (N - m)

Work is scalar (not vector).

Sign conventions:

• Positive: force has component in direction of displacement
• Negative: force has component opposite to displacement
• Zero: force perpendicular to displacement

Examples:

• Weight does positive work falling, negative work rising
• Normal force does zero work (perpendicular to surface)
• Tension does zero work if motion perpendicular to rope

Work from variable force (force depends on position).

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

Vector form: $W = \int \vec{F} \cdot d\vec{r}$

Work from a Force vs. Position Graph:

Work equals area under force-position graph.

$W = \text{Area under F vs. x curve}$

Properties:

• Above x-axis: positive work
• Below x-axis: negative work
• Net work = signed area

Work equals area under force-position graph.

$W = \text{Area under F vs. x curve}$

Properties:

• Above x-axis: positive work
• Below x-axis: negative work
• Net work = signed area

Net work equals change in kinetic energy.

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

$W_{net} = \frac{1}{2}mv_f^2 - \frac{1}{2}mv_i^2$

Physical meaning:

• Work transfers energy to/from object as kinetic energy
• Applies for any forces (conservative or non)
• Alternative to Newton's Second Law integration

Work done against gravity near Earth's surface.

Reference: Choose U = 0 at some reference height.

Near Earth (constant g): $U_g = mgh$

Where h = height above reference level.

Universal gravitation: $U_g = -\frac{Gm_1 m_2}{r}$

Reference: U = 0 at r -> ∞

Work done against spring force.

Hooke's Law: $F = -kx$

Gravitational PE near Earth's surface.

$U_g = mgh$

Properties:

• Positive for h > reference
• Increases with height
• Reference choice arbitrary (only DeltaU matters)

U_spring = (1/2)kx2:

Elastic potential energy of spring.

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

Properties:

• Always positive (depends on x2)
• Maximum at maximum compression/extension
• Zero at equilibrium (x = 0)

Conservative forces (work independent of path):

• Gravity
• Spring force
• Electric force

Non-conservative forces (work depends on path):

• Friction
• Air resistance
• Push/pull forces

Test: $\oint \vec{F} \cdot d\vec{l} = 0$ around any closed path

Total mechanical energy constant for conservative forces.

$E_{mech} = K + U = \text{constant}$

$\frac{1}{2}mv_i^2 + mgh_i + \frac{1}{2}kx_i^2 = \frac{1}{2}mv_f^2 + mgh_f + \frac{1}{2}kx_f^2$

When applicable:

• Only conservative forces
• Isolated system
• No energy input/output

With non-conservative forces: $W_{nc} = \Delta E_{mech} = \Delta K + \Delta U$

General form including non-conservative work.

$W_{nc} = \Delta E_{mech} = \Delta K + \Delta U$

Where $W_{nc}$ = work done by non-conservative forces (friction, external pushes)

Explicit form: $W_{friction} + W_{external} = \left(\frac{1}{2}mv_f^2 - \frac{1}{2}mv_i^2\right) + (U_f - U_i)$

Applications:

• Problems with friction
• Energy with external inputs/outputs
• Non-ideal situations

P_avg = W/Deltat:

Average power is work per unit time.

$P_{avg} = \frac{W}{\Delta t}$

Units: Watts (W) = Joules/second (J/s)

Physical meaning:

• Rate of energy transfer
• Force component in direction of motion times speed

Instantaneous power.

$P = \vec{F} \cdot \vec{v} = Fv\cos\theta$

Where:

• $\vec{F}$ = force
• $\vec{v}$ = velocity at that instant

Physical meaning:

• Rate of energy transfer
• Force component in direction of motion times speed

Relationships: $P = Fv = \frac{W}{t} = \frac{dE}{dt}$