Physics · Units 1–10 · Quick reference

Physics 1

Grades 10–12+8 pages

Units, vectors and significant figures; motion and projectiles; Newton’s laws, friction and pulleys; circular motion and gravity; energy, momentum and rotation; oscillations, waves and sound; fluids; heat; electricity, magnetism and optics.

1Measurement2Constants31D motion4Projectiles5Newton6Friction7Circles & gravity8Energy9Momentum10Rotation11Oscillations12Waves & sound13Fluids14Heat15Electricity16Magnetism17Optics

1Measurement, Units & Vectors

The seven SI base units
Quantity	Unit	Symbol
length	meter	m
mass	kilogram	kg
time	second	s
electric current	ampere	A
temperature	kelvin	K
amount	mole	mol
brightness	candela	cd

Metric prefixes

G, M, k: 10⁹, 10⁶, 10³ (giga, mega, kilo)
c, m: 10⁻², 10⁻³ (centi, milli)
μ, n: 10⁻⁶, 10⁻⁹ (micro, nano)

Gentle Mice Keep Cheese Mostly Under Nightstands

Prefixes from big to small: G, M, k, c, m, μ, n.

Derived units
Unit	Equals	Unit	Equals
N	kg·m/s²	Pa	N/m²
J	N·m	Hz	1/s
W	J/s	C	A·s
V	J/C	Ω	V/A

Convert with unit fractions Worked

65 mi/h × 1609 m1 mi × 1 h3600 sunits cancel

= 29.05 m/s ≈ 29 m/s2 significant figures

Significant figures

Nonzero digits and zeros between them count; leading zeros never do (0.00340 has 3).
Trailing zeros count only with a decimal point (2.50 has 3).
× and ÷: fewest significant figures. + and −: fewest decimal places.
Counted and defined numbers are exact.

Rounding the answer Worked

12.0 cm × 3.5 cm = 42 cm²not 42.00

12.11 m + 0.3 m = 12.4 mtenths place

Resolve a vector into perpendicular components.

Components and resultant

Scalar: size only (12 s). Vector: size and direction (5 m north). Add x parts and y parts separately.

Aₓ = A cos θ, A_y = A sin θ

R = √(Rₓ² + R_y²)

θ = tan⁻¹(R_y ÷ Rₓ)

Walk east, then north Worked

3.0 km E + 4.0 km N: R = √(3.0² + 4.0²) = 5.0 km

θ = tan⁻¹(4.03.0) = 53° north of east

Cos is close

The cosine goes with the side close to the angle, so with θ measured from the x-axis, A cos θ is the x part.

Honest measure

Accurate: near the true value. Precise: repeats agree.
Percent error = |measured − accepted| ÷ accepted × 100 %.
“A false balance is abomination to the LORD: but a just weight is his delight” (Proverbs 11:1, KJV).

2Constants & Key Equations

Physical constants
Constant	Value
g (Earth’s surface)	9.8 m/s²
G (gravitation)	6.674 × 10⁻¹¹ N·m²/kg²
k (Coulomb)	8.99 × 10⁹ N·m²/C²
e (elementary charge)	1.602 × 10⁻¹⁹ C
c (light in vacuum)	3.00 × 10⁸ m/s
μ₀ (magnetic)	4π × 10⁻⁷ T·m/A
Earth: mass, radius	5.97 × 10²⁴ kg, 6371 km

Solving any problem

Sketch; list knowns with units; name the unknown.
Pick an equation where only the unknown is missing; solve with symbols, then numbers.
Check units, size and sig figs.

Handy values
Quantity	Value
water density	1000 kg/m³
water specific heat	4186 J/(kg·°C)
ice melts (L_f)	334 kJ/kg
water boils (L_v)	2260 kJ/kg
1 atm	101.3 kPa = 760 mm Hg
sound in air, 20 °C	343 m/s
n: air, water, glass	1.00, 1.33, ≈ 1.50
absolute zero	0 K = −273.15 °C
1 cal; 1 hp	4.186 J; 746 W
1 kWh	3.6 × 10⁶ J

Units ride along

Carry units through every line; wrong units in the answer mean an algebra slip.

Key equations
Topic	Equations
Motion	v = v₀ + at; Δx = v₀t + ½at²; v² = v₀² + 2aΔx
Forces	F_net = ma; W = mg; f = μN; F = kx
Gravity	a_c = v²/r; F = Gm₁m₂/r²
Energy	W = Fd cos θ; KE = ½mv²; PE = mgh; P = W/t
Momentum	p = mv; FΔt = Δp
Rotation	τ = rF sin θ; τ = Iα; L = Iω
Waves	v = fλ; T = 2π√(L/g); T = 2π√(m/k)
Fluids	P = F/A; P = P₀ + ρgh; F_b = ρVg
Heat	Q = mcΔT; Q = mL; ΔU = Q − W
Electricity	F = kq₁q₂/r²; V = IR; P = IV
Magnetism	F = BIL sin θ; F = qvB sin θ
Optics	n₁ sin θ₁ = n₂ sin θ₂; 1/f = 1/d_o + 1/d_i

Physics 1Physics · Units 1–10 · continued

3Motion in One Dimension

Displacement Δx: change in position, with direction (vector)
Distance: total path length (scalar)
Velocity v: Δx ÷ Δt, with direction
Speed: size of velocity
Acceleration a: Δv ÷ Δt (m/s²): speeding up, slowing down or turning

The kinematic equations
Equation (constant a)	Leaves out
v = v₀ + at	Δx
Δx = v₀t + ½at²	v
v² = v₀² + 2aΔx	t
Δx = ½(v₀ + v)t	a

Skip the stranger

List the five letters v₀, v, a, t, Δx. Choose the equation that leaves out the one you neither know nor want.

Speeding up Worked

car: 0 → 27 m/s in 9.0 s
a = 27 − 09.0 = 3.0 m/s²
Δx = ½(0 + 27)(9.0) = 121.5 ≈ 120 m

Braking distance Worked

25 m/s, a = −5.0 m/s², to rest
Δx = 0² − 25²2(−5.0) = 62.5 mv² = v₀² + 2aΔx
t = 0 − 25−5.0 = 5.0 s

Free fall

Take up as +: a = −g = −9.8 m/s², going up or coming down.
Without air, all objects fall alike (Apollo 15 hammer and feather, 1971).
At the top v = 0 but a is still −g.

Thrown straight up at 19.6 m/s Worked

top: t = 19.69.8 = 2.0 sv = 0

height = 19.6²2(9.8) = 19.6 mback down at 4.0 s

Steady acceleration from rest: the x–t curve steepens, v–t is a straight line, a–t is flat.

Dropped from 45 m Worked

t = √(2hg) = √(2(45)9.8) = 3.0 s

v = gt = (9.8)(3.03) = 30 m/s

Reading a v–t graph Worked

v rises 0 → 12 m/s in 4.0 s, then holds 3.0 s
a = 124.0 = 3.0 m/s²slope
Δx = ½(4.0)(12) + (12)(3.0) = 60 marea

Reading motion graphs
Graph	Slope gives	Area gives
x–t	velocity	—
v–t	acceleration	displacement
a–t	—	change in v

Round trip Worked

60 km out at 60 km/h, back at 30 km/h
average speed = 120 km1 h + 2 h = 40 km/hnot 45
average velocity = 0back where it started

Watch out

Negative a does not always mean slowing; it means a points in the − direction.
A flat x–t line means at rest; a flat v–t line means steady velocity.

4Projectiles

Horizontal velocity stays the same; vertical velocity shrinks to 0 at the top, then grows downward.

The key idea

Horizontal and vertical motions are independent; time links them.
Horizontal: no force (ignoring air), so vₓ stays constant.
Vertical: free fall, a = −g.

Galileo showed the path is a parabola (Two New Sciences, 1638).

Off a cliff Worked

15 m/s horizontally from 20 m up
fall time t = √(2(20)9.8) = 2.02 ssame as a drop
x = (15)(2.02) = 30 m

Projectile equations

R and H formulas: level ground, no air.

vₓ = v₀ cos θ, v_0y = v₀ sin θ

x = vₓt, y = v_0yt − ½gt²

R = v₀² sin 2θg, H = (v₀ sin θ)²2g

Launched at an angle Worked

20 m/s at 30° on level ground
vₓ = 17.3 m/s, v0y = 10.0 m/s
time aloft = 2(10.0)9.8 = 2.04 s
R = 20² sin 60°9.8 = 35.3 m; H = 10.0²2(9.8) = 5.10 m

Watch out

At the top the velocity is vₓ, not zero.
The acceleration is g downward at every point of the flight.

Facts worth knowing

On level ground 45° gives the longest range (no air).
Angles that add to 90° (30° and 60°) give the same range.
A ball dropped and one thrown sideways from the same height land together.

Crossing a river Worked

boat 4.0 m/s straight across; current 3.0 m/s; river 80 m wide
speed over ground √(4.0² + 3.0²) = 5.0 m/s, 37° downstream
time 804.0 = 20 s; drift (3.0)(20) = 60 mvelocities add as vectors

Ruler across, stone down

Sideways the motion is steady as a ruler; up and down it falls like a dropped stone. Solve each, then join them by t.

Physics 1Physics · Units 1–10 · continued

5Newton’s Laws & Free-Body Diagrams

Newton’s three laws (Principia, 1687)
Law	In words	Example
1st (inertia)	An object keeps its velocity (or rest) unless a net force acts.	riders lurch forward when a car stops
2nd	F_net = ma: acceleration follows the net force.	twice the force, twice the acceleration
3rd	Forces come in equal, opposite pairs on two different objects.	a rocket pushes gas down; gas pushes it up

Force: a push or pull, in newtons (N)
Mass m: amount of matter; measures inertia (kg)
Weight W: gravity’s pull: W = mg (N)
Equilibrium: ΣF = 0: at rest or at constant velocity

Drawing a free-body diagram

Draw the object alone as a box or dot.
Draw every force on it as an arrow from it; label each.
Choose axes, one along the acceleration.
Write ΣFₓ = maₓ and ΣF_y = ma_y; solve.

Forces to look for
Force	Direction
Weight W = mg	straight down
Normal N	perpendicular out of a surface
Tension T	along a rope, pulling
Friction f	along a surface, against sliding
Spring F = kx	back toward rest length (Hooke, 1678)
Air drag	against the motion

F = ma Worked

1200 kg car, net force 3600 N

a = 36001200 = 3.0 m/s²

Riding an elevator Worked

60 kg person, a = 2.0 m/s² up
N − mg = ma → N = 60(9.8 + 2.0) = 708 Nfeels heavier
speeding up downward: N = 60(9.8 − 2.0) = 468 Nfeels lighter

Weight on the Moon Worked

60 kg astronaut; Moon g = 1.62 m/s²

Earth: (60)(9.8) = 588 N; Moon: (60)(1.62) = 97 Nsame 60 kg of mass

Hanging sign Worked

20 kg sign, two cables each 30° above level

2T sin 30° = mg → T = (20)(9.8)2(0.500) = 196 Neach cable

Terminal velocity

Air drag grows with speed until it equals the weight; then ΣF = 0 and the speed stays constant.

Two objects, two forces

A third-law pair acts on two different objects, so the pair never cancels. Forces that cancel act on the same object.

6Friction, Inclines & Pulleys

Block on an incline: weight splits into mg sin θ down the slope and mg cos θ into it; friction points up the slope against sliding.

Friction

Static friction grows to match the push, up to its limit; usually μ_s > μ_k. Slipping angle: μ_s = tan θ.

f_s ≤ μ_sN, f_k = μ_kN

Incline (angle θ)

F_∥ = mg sin θ, N = mg cos θ

a = g sin θ (no friction)

a = g(sin θ − μ_k cos θ) (sliding down)

Typical coefficients (approximate)
Surfaces	μ_s	μ_k
rubber on dry concrete	1.0	0.8
wood on wood	0.25–0.5	0.2
steel on steel (dry)	0.74	0.57
ice on ice	0.1	0.03

Pushing a crate Worked

20 kg, μk = 0.30, push 100 N level
f = 0.30(20)(9.8) = 58.8 N
a = 100 − 58.820 = 2.06 m/s²

Sliding down a ramp Worked

30°, μ_k = 0.20

a = 9.8(0.500 − 0.20 × 0.866) = 3.20 m/s²mass cancels

Atwood machine: the heavier side falls.

Atwood machine

a = (m₂ − m₁)gm₁ + m₂, T = 2m₁m₂gm₁ + m₂

3.0 kg and 5.0 kg Worked

a = (5.0 − 3.0)(9.8)8.0 = 2.45 m/s²

T = 2(3.0)(5.0)(9.8)8.0 = 36.8 Nbetween the two weights

Two blocks pulled together Worked

30 N pulls 2.0 kg + 4.0 kg on a frictionless floor
a = 306.0 = 5.0 m/s²treat them as one
rope between them pulls the 2.0 kg: T = (2.0)(5.0) = 10 Nif the 30 N pulls the 4.0 kg

Friction lies down, normal stands up

Friction runs along the surface against sliding; the normal force stands perpendicular to it. On a slope, N is less than mg.

Physics 1Physics · Units 1–10 · continued

7Circular Motion & Gravitation

Velocity is tangent; acceleration and net force point to the center.

Uniform circular motion

a_c = v²r = 4π²rT²

F_c = mv²r, v = 2πrT

What supplies the force?

Centripetal force is not a new force: it is the net inward force from tension, gravity, friction or the normal force.
The outward “pull” riders feel is their own inertia.

Earth’s g from G Worked

g = (6.674 × 10⁻¹¹)(5.97 × 10²⁴)(6.371 × 10⁶)²

= 9.82 m/s²matches 9.8

Car on a flat curve Worked

1000 kg at 20 m/s, r = 50 m
ac = 20²50 = 8.0 m/s²; F = 8000 Nfrom friction
needs μs ≥ 8.09.8 = 0.82

Universal gravitation

r is center to center. Cavendish measured G with a torsion balance (1798).

F = Gm₁m₂r², g = GMr²

Twice as far, a quarter the pull

Gravity and the electric force fall off as 1/r²: double r and the force drops to ¼; triple it, to 1/9.

Kepler’s laws of planets
Kepler’s law	Says
1st (1609)	Planets move in ellipses with the Sun at one focus.
2nd (1609)	A line to the Sun sweeps equal areas in equal times: faster when closer.
3rd (1619)	T² ∝ a³; around the Sun, T² = a³ (T in years, a in AU).

Circular orbits

GM_E = 3.986 × 10¹⁴ m³/s².

v = √(GMr), T = 2π√(r³GM)

Space station orbit Worked

410 km up: r = 6371 + 410 = 6781 km
v = √(3.986 × 10¹⁴6.781 × 10⁶) = 7.67 km/s
T = 2πrv = 5557 s ≈ 93 min

Loop-the-loop Worked

loop radius r = 10 m
at the top gravity alone supplies Fc: v ≥ √(gr) = 9.9 m/s
frictionless start height ≥ 2.5r = 25 m

Kepler’s third law Worked

Mars: a = 1.52 AU

T = √(1.52³) = 1.87 years

Held in place

“He stretcheth out the north over the empty place, and hangeth the earth upon nothing” (Job 26:7, KJV).

8Work, Energy & Power

Work

In joules. θ is between the force and the motion. A force at 90° to the motion does no work.

W = Fd cos θ

Pulling a wagon Worked

50 N at 30° above level, 10 m

W = (50)(10) cos 30° = 433 J

Work–energy theorem

W_net = ΔKE = ½mv_f² − ½mv_i²

Roller coaster drop Worked

from rest, 20 m down, no friction

mgh = ½mv² → v = √(2gh) = √(2(9.8)(20)) = 19.8 m/smass cancels

Kinds of energy
Energy	Formula
Kinetic	KE = ½mv²
Gravitational potential	PE = mgh
Spring (elastic)	PE = ½kx²
Thermal, chemical, electrical	stored or moving in particles and fields

Conservation of energy

Without friction: KE_i + PE_i = KE_f + PE_f.
With friction, the lost mechanical energy becomes thermal energy; the total stays the same.

Joule’s falling-weight paddle wheel (1845) showed heat is a form of energy.

Spring launcher Worked

k = 200 N/m pressed 0.10 m
PE = ½(200)(0.10)² = 1.0 J
0.050 kg ball: v = √(2(1.0)0.050) = 6.3 m/s

Power

In watts (J/s). James Watt’s unit: 1 hp = 746 W. Energy bills use 1 kWh = 3.6 × 10⁶ J.

P = Wt = Fv

Climbing stairs Worked

60 kg rises 3.0 m in 4.0 s

P = (60)(9.8)(3.0)4.0 = 441 W ≈ 0.59 hp

Efficiency: useful energy outenergy in × 100 %; always under 100 %
Kilowatt-hour: a 100 W bulb lit for 10 h uses 1 kWh

Push along the path

Work needs force along the motion. Carrying a tray across a level room, or gravity on a circular orbit, does zero work.

Physics 1Physics · Units 1–10 · continued

9Momentum & Collisions

Momentum and impulse

In kg·m/s (= N·s). Momentum is a vector: pick a + direction.

p = mv

J = F_avgΔt = Δp

Batting a ball Worked

0.15 kg ball: +40 m/s in, −50 m/s out; contact 1.5 ms
Δp = 0.15(−50 − 40) = −13.5 kg·m/s
F = 13.50.0015 = 9000 N

Stretch the time

The same Δp over a longer time means a smaller force: air bags, seat belts, crumple zones, bent knees on landing.

Conservation of momentum

Holds whenever no outside net force acts on the system.

m₁v₁ + m₂v₂ = m₁v₁′ + m₂v₂′

Truck meets car; they stick Worked

2000 kg at 10 m/s hits 1000 kg at rest
v′ = (2000)(10)3000 = 6.67 m/s
KE: 100 kJ → 66.7 kJ; ⅓ lost

Kinds of collisions
Collision	Momentum	Kinetic energy
Elastic	conserved	conserved (equal billiard balls swap speeds)
Inelastic	conserved	some lost to heat, sound, bending
Perfectly inelastic	conserved	most lost: objects stick

Add up before, add up after

Total momentum before equals total after. Choose + once, give each velocity its sign, then solve.

10Rotation, Torque & Balance

Angle θ: in radians: 1 rev = 2π rad = 360°
Angular velocity ω: Δθ ÷ Δt (rad/s); v = ωr
Angular acceleration α: Δω ÷ Δt (rad/s²)

Balanced when the torques match: (30)(9.8)(2.0) = (40)(9.8)(1.5).

Equilibrium

An object tips once its center of mass passes outside its base.

ΣF = 0 and Στ = 0

F₁d₁ = F₂d₂ (lever)

Torque

Lever arm: the perpendicular distance from the pivot to the force’s line.

τ = rF sin θ = F × lever arm

Wrench Worked

40 N, 0.30 m from the bolt
at 90°: τ = (0.30)(40) = 12 N·m
at 30°: τ = 12 sin 30° = 6.0 N·m

Each linear idea has a turning twin
Linear	Rotational
x, v, a	θ, ω, α
mass m	rotational inertia I
F = ma	τ = Iα
KE = ½mv²	KE = ½Iω²
p = mv	L = Iω

Rotational inertia
Shape (mass M, radius R)	I
hoop or ring	MR²
solid disk	½MR²
solid sphere	⅖MR²
rod (length L) about center	112ML²
rod about one end	⅓ML²

Mass far from the axis makes I larger: a hoop rolls downhill slower than a disk.

Spinning skater Worked

arms in: I drops from 4.0 to 2.0 kg·m²

I₁ω₁ = I₂ω₂: 2.0 rev/s → 4.0 rev/sL conserved

Leverage is length

The same push gives more torque farther from the pivot: grab a door at the handle, not near the hinge, and push at right angles.

11Oscillations & Pendulums

Period T: time for one full cycle (s)
Frequency f: cycles per second, f = 1/T (Hz)
Amplitude A: largest distance from rest; restoring force F = −kx

Periods

Pendulum: small swings; mass and amplitude do not matter.

pendulum: T = 2π√(Lg)

spring: T = 2π√(mk)

A pendulum and a mass on a spring: both are simple harmonic motion.

Four times the string, twice the swing

T grows with √L: make a pendulum 4× longer and its period doubles.

Two periods Worked

1.00 m pendulum: T = 2π√(1.009.8) = 2.01 s≈ 1 s each way

0.50 kg on k = 200 N/m: T = 2π√(0.50200) = 0.314 s; f = 3.18 Hz

Energy in SHM

Ends: speed 0; PE and acceleration greatest. Center: speed and KE greatest.
Huygens built the first pendulum clock (1656).

Physics 1Physics · Units 1–10 · continued

12Waves & Sound

Transverse waves shake across the travel direction; longitudinal waves (sound) shake along it.

Mechanical wave: needs a medium (sound, water, string)
Electromagnetic wave: needs none (light, radio)
Wavelength λ: crest to crest (m)

Wave speed

The medium sets v; the source sets f.

v = fλ

What waves do

Reflect (echo), refract (bend entering a new medium), diffract (spread around edges).
Interfere: in step add (constructive); out of step cancel (destructive).
Two waves crossing back and forth make standing waves with nodes (still) and antinodes.

Concert A Worked

440 Hz in air at 343 m/s

λ = 343440 = 0.780 m

Standing waves

String fixed at both ends, n = 1, 2, 3, … Tube closed at one end: λ₁ = 4L.

λₙ = 2Ln, fₙ = nv2L

Sound in brief
Sound	Fact
speed	≈ (331 + 0.6T) m/s; 343 m/s at 20 °C
pitch	set by frequency
loudness	set by amplitude; +10 dB = 10× the intensity
human range	about 20 to 20,000 Hz

Light is a wave too

Young’s two-slit experiment (early 1800s): bright fringes where waves from the slits arrive in step.

d sin θ = mλ, Δy = λLd

Guitar string Worked

L = 0.65 m, v = 286 m/s

f₁ = 2862(0.65) = 220 Hz; f₂ = 440 Hz

Red laser, two slits Worked

λ = 650 nm, d = 0.20 mm, screen L = 2.0 m

Δy = (650 × 10⁻⁹)(2.0)0.20 × 10⁻³ = 6.5 mmfringe spacing

Doppler effect

Upper signs when moving toward each other. Doppler (1842); Buys Ballot tested it with horn players on a train (1845).

f′ = f v ± v_ov ∓ v_s

Passing horn Worked

500 Hz horn, 25 m/s, still listener
coming: 500 × 343343 − 25 = 539 Hz
going: 500 × 343343 + 25 = 466 Hz

Coming close, pitch goes up

Waves bunch up ahead of a moving source (higher f) and spread out behind it (lower f).

13Fluids

Density and pressure

Pressure in pascals (N/m²) acts in every direction and grows with depth.

ρ = mV, P = FA

P = P₀ + ρgh

Ten meters down Worked

P = 101.3 kPa + (1000)(9.8)(10) Pa

= 101.3 + 98 = 199 kPa ≈ 2 atm

Pascal’s principle

Pressure on a confined fluid passes undiminished to every part: F₁A₁ = F₂A₂.

At Pascal’s request, Périer carried a barometer up the Puy de Dôme (1648); the mercury fell with height.

Ten meters, one atmosphere

Each 10 m of water adds about one atmosphere of pressure.

Hydraulic lift Worked

200 N on a 0.010 m² piston; big piston 0.50 m²

F₂ = 200 × 0.500.010 = 10,000 Nsmall force, long stroke

Archimedes’ principle

Buoyant force = weight of the fluid pushed aside. Floats if ρ_object < ρ_fluid; fraction under = ρ_object ÷ ρ_fluid.

F_b = ρ_fluidV_subg

Rock under water Worked

V = 0.0020 m³, ρ = 2650 kg/m³
Fb = (1000)(0.0020)(9.8) = 19.6 N
weight 51.9 N; in water it weighs 32.3 N

Iceberg Worked

ice 917 kg/m³ in seawater 1025 kg/m³

9171025 = 89 % under the surface

Moving fluids

Continuity, and Bernoulli’s equation (Hydrodynamica, 1738): along a flow, faster fluid has lower pressure.

A₁v₁ = A₂v₂

P + ½ρv² + ρgh = constant

Leaking tank Worked

hole 0.80 m below the water surface

v = √(2gh) = √(2(9.8)(0.80)) = 4.0 m/sTorricelli

Thumb on the hose Worked

cut the opening to ¼ of its area

v₂ = v₁ × A₁A₂ = 4× faster

Physics 1Physics · Units 1–10 · continued

14Heat & Thermodynamics

Temperature: average kinetic energy of the particles
Heat Q: energy that flows because of a temperature difference
Specific heat c: J to warm 1 kg by 1 °C
Latent heat L: J per kg to melt or boil, with no temperature change

Heating water: temperature climbs, then holds flat while ice melts and while water boils.

Laws of thermodynamics
Law	Says
Zeroth	Two objects in balance with a third are in balance with each other.
First	ΔU = Q − W: energy is conserved (500 J in, 200 J of work out: ΔU = +300 J).
Second	Heat flows hot → cold on its own; no engine turns all heat into work; entropy grows.
Third	Absolute zero cannot be reached.

Scales

T_K = T_C + 273.15

T_F = 95T_C + 32

EX25 °C = 298 K = 77 °F. Kelvin set the absolute scale in 1848.

Heat equations

Q = mcΔT, Q = mL

Boiling water for pasta Worked

2.0 kg water, 20 → 100 °C

Q = (2.0)(4186)(80) = 669,760 J ≈ 670 kJ

Melting ice Worked

0.50 kg at 0 °C

Q = (0.50)(334 kJ/kg) = 167 kJtemperature holds at 0 °C

Mixing water Worked

0.20 kg at 80 °C + 0.30 kg at 20 °Cheat lost = heat gained

T = (0.20)(80) + (0.30)(20)0.50 = 44 °C

Three ways heat moves
Transfer	How	Example
Conduction	contact, particle to particle	hot pan handle
Convection	warm fluid rises and circulates	boiling pot, sea breeze
Radiation	electromagnetic waves; no medium	sunlight, campfire

Heat engines

The best possible engine: Carnot (1824).

e = WQ_H = 1 − Q_CQ_H

e_max = 1 − T_CT_H (kelvins)

Best possible engine Worked

T_H = 500 K, T_C = 300 K

e_max = 1 − 300500 = 40 %

Ratios need kelvins

Use kelvins whenever temperatures are multiplied or divided (efficiency, gas laws). A change of 1 °C equals a change of 1 K.

15Electricity: Charge & Circuits

Electric charge

Two kinds: + and −. Like charges repel; opposites attract.
Charge is conserved and comes in whole multiples of e = 1.602 × 10⁻¹⁹ C.
Objects charge by friction, contact or induction; conductors let charge move, insulators do not.

Coulomb’s law

k = 8.99 × 10⁹ N·m²/C². Coulomb measured it with a torsion balance (1785).

F = kq₁q₂r²

Current I: charge per second, q ÷ t (amperes)
Voltage V: energy per charge (J/C = volts)
Resistance R: opposition to current (ohms)
Conventional current: flows + to −; electrons drift the other way

Two charged spheres Worked

+2.0 μC and +2.0 μC, 0.30 m apart

F = (8.99 × 10⁹)(2.0 × 10⁻⁶)²0.30² = 0.40 Nrepel

Ohm’s law and power

Ohm (1827). Resistance of a wire: R = ρL ÷ A.

V = IR

P = IV = I²R = V²R

Series circuit Worked

2 Ω + 4 Ω + 6 Ω on 12 V
R = 12 Ω, I = 1212 = 1.0 A
drops: 2 V, 4 V, 6 V (sum 12 V)

Series shares volts, parallel shares amps

In series the voltage divides and current is the same; in parallel the current divides and voltage is the same.

The two ways to connect resistors (values used in the examples below).

Series vs parallel
	Series	Parallel
Current	same everywhere	splits among branches
Voltage	shared (drops add up)	same across each branch
Total R	R₁ + R₂ + …	1R = 1R₁ + 1R₂ + …
One bulb out	all go out	others stay lit (house wiring)

Parallel circuit Worked

3 Ω and 6 Ω on 12 V
R = (3)(6)3 + 6 = 2.0 Ω; I = 6.0 A
branches: 4.0 A and 2.0 A

Physics 1Physics · Units 1–10 · continued

16Magnetism

Field lines leave the north pole and loop around into the south pole.

Right-hand grip

Thumb along the current, fingers curl the way B circles the wire. (Traditional.)

Magnetic force

On a current-carrying wire, and on a moving charge. B in tesla (T); the force is perpendicular to both B and the motion.

F = BIL sin θ

F = qvB sin θ

Magnets

Every magnet has a north and a south pole; like poles repel, unlike attract. Cutting a magnet makes two smaller magnets.
Earth is a giant magnet (Gilbert, De Magnete, 1600); a compass’s north end points toward geographic north.

Currents make magnetism

Oersted saw a current turn a compass needle (1820).
Straight wire: B = μ₀I2πr; coil (solenoid): B = μ₀nI.

Induction

A changing magnetic field through a coil makes a voltage (Faraday, 1831): generators turn motion into electricity.
Transformer: V_sV_p = N_sN_p.

Force on a wire Worked

0.50 T, 2.0 A, 0.30 m, at 90°

F = (0.50)(2.0)(0.30) = 0.30 N

Field near a wire Worked

10 A, 2.0 cm away

B = (4π × 10⁻⁷)(10)2π(0.020) = 1.0 × 10⁻⁴ Tabout twice Earth’s field

Step-down transformer Worked

120 V, 500 turns in, 50 turns out

V_s = 120 × 50500 = 12 V

Go deeper

Loops and solenoids, electromagnets, the motor effect, DC motors, and Faraday’s and Lenz’s laws: see the School Launch Magnetism and Induction sheet (Physics, Unit 10).

17Light & Optics

Light: an electromagnetic wave; c = 3.00 × 10⁸ m/s in vacuum
Spectrum: radio, microwave, infrared, visible (700–400 nm), ultraviolet, X-ray, gamma: rising frequency
Index n: n = c ÷ v: how much a material slows light

ROY G BIV

Red, orange, yellow, green, blue, indigo, violet: longest to shortest visible wavelength. (Traditional.)

Reflection

Angle of reflection = angle of incidence, both from the normal.
A flat mirror makes an upright, same-size, virtual image as far behind as the object is in front.

Entering a slower medium, light bends toward the normal.

Refraction (Snell’s law)

Snell (1621); Ibn Sahl found it about 984. Past the critical angle θ_c, light reflects totally (optical fibers).

n₁ sin θ₁ = n₂ sin θ₂

sin θ_c = n₂n₁

Air into water Worked

30° from the normal, n = 1.33
sin θ₂ = sin 30°1.33 → θ₂ = 22.1°
speed in water: 3.00 × 10⁸1.33 = 2.26 × 10⁸ m/s

Critical angle Worked

glass (1.50) to air

θ_c = sin⁻¹(1.001.50) = 41.8°

Where the image lands
Object (converging lens)	Image
beyond 2F	real, inverted, smaller (camera)
between F and 2F	real, inverted, larger (projector)
inside F	virtual, upright, larger (magnifier)
any (diverging lens)	virtual, upright, smaller

Converging lens, f = 10 cm, object at 30 cm: three rays meet at a real, inverted image.

Thin lens and mirror

f + for converging lenses and concave mirrors; d_i + real, − virtual; m − inverted.

1f = 1d_o + 1d_i

m = −d_id_o = h_ih_o

Camera-style image Worked

f = 10 cm, d_o = 30 cm: 1d_i = 110 − 130 → d_i = 15 cm

m = −1530 = −0.50: real, inverted, half size

Real images flip

A real image is inverted and can fall on a screen; a virtual image is upright and only seen by looking through the lens or into the mirror.