Learning Physics

Mirror Equation

The mirror formula relates the focal length of a spherical mirror to the object and image distances. Move the sliders to explore how convex and concave mirrors form real and virtual images.

1⁄_f  =  1⁄_u  +  1⁄_v

The canvas traces the principal axis, focal markers, and image position so the geometry stays tied to the algebra.

u 28.0

f -21.0

¹⁄_f = ¹⁄_u + ¹⁄_v, v ≈ −12, m ≈ 0.43

Ideal Gas Law (PV = nRT)

Lock one variable and adjust the others to see how pressure, volume, amount, and temperature are related. The particle simulation shows molecules bouncing inside a cylinder whose height scales with volume.

PV = nRT

Keeping one quantity fixed makes the compensating changes in the remaining three easier to read as both a formula and a physical picture.

P 1.00

V 24.5

n 1.00

T 298.0

Locked n: move P, V, or T to solve for the amount of gas.

PV and nRT stay matched while the locked variable updates.

Charles’ Law

Charles’ law says the volume of a gas is proportional to its absolute temperature when pressure and amount of gas stay fixed. Heating the gas makes the particles occupy more space, so the container must expand if pressure is to remain unchanged.

V ∝ T V₁⁄T₁ = V₂⁄T₂

The law is linear only when temperature is measured from absolute zero. That is why the formula uses kelvin rather than Celsius.

Coulomb’s Law

Coulomb’s law gives the electrostatic force between two point charges. The interaction acts along the line joining the charges, and its size depends on the product of the charge magnitudes.

F = k|q₁q₂|⁄r²

Distance has a strong effect: doubling the separation reduces the force to one quarter. The sign of the charges determines whether the force pulls the particles together or pushes them apart.

Degrees of Freedom

Degrees of freedom count the independent coordinates or energy modes needed to describe a system. In kinetic theory, they determine how thermal energy is distributed among translation, rotation, and sometimes vibration.

U = f⁄2 nRT ⟨E⟩ = f⁄2 k_BT

This is why monatomic and diatomic gases can have different heat capacities even at the same temperature: they do not have the same number of active modes.

Hooke’s Law

Hooke’s law describes the restoring force of an ideal spring for small displacements. Pull the spring farther from equilibrium and the opposing force grows in direct proportion.

F = -kx

The relationship is linear only within the elastic regime. Beyond that range, real materials stop behaving like ideal springs.

Kinetic Energy

Kinetic energy is the energy an object has because it is moving. The formula shows that speed matters more strongly than mass, since velocity is squared.

K = 1⁄2 mv²

This is why modest increases in speed can create large increases in impact or required stopping distance. The work-energy theorem ties that change directly to net work.

Lens Equation

The thin-lens equation links focal length, object distance, and image distance. It predicts where an image forms when light passes through a converging or diverging lens.

1⁄f = 1⁄d_o + 1⁄d_i

Like the mirror equation, it turns a geometric ray diagram into a compact algebraic rule. Changing sign conventions changes the bookkeeping, not the underlying optics.

Ohm’s Law

Ohm’s law connects voltage, current, and resistance in simple electrical circuits. It is one of the basic translation rules between how hard charges are pushed and how much current actually flows.

V = IR

The law is an idealized model for ohmic materials. Real devices such as diodes and lamps can depart from this linear behavior.

Period-Frequency Relation

Period and frequency describe the same repeating process from opposite viewpoints. One asks how long a single cycle lasts; the other asks how many cycles happen each second.

f = 1⁄_T,   T = 1⁄_f

Because they are reciprocals, increasing one necessarily decreases the other. This relation appears everywhere from waves and circuits to clocks and orbital motion.

Potential Energy

Potential energy is stored energy associated with position or configuration. In the near-Earth gravitational model, lifting an object higher increases its potential energy by an amount proportional to height.

U = mgh

This is the energy reserve that can later become kinetic energy as the object falls. Different systems use different potential formulas, but the idea of stored capacity for work is the same.

Newton’s Second Law

Newton’s second law links force to acceleration through mass. Push harder and acceleration increases; keep the same force but increase mass and the acceleration drops.

F = ma

The equation explains both why light objects respond quickly and why heavy ones resist changes in motion. Force sets the rate at which velocity changes.

Wave Speed Relation

Wave speed connects how often crests pass by with how far apart they are. If the medium keeps the wave speed fixed, raising frequency forces wavelength to shrink.

v = fλ

This simple product ties together sound, light, strings, and water waves. It explains why frequency and wavelength trade off against each other.

Snell’s Law

Snell’s law describes how light bends when it passes between materials with different refractive indices. The change in angle reflects the change in propagation speed.

n₁ sin θ₁ = n₂ sin θ₂

When light enters a slower medium, it bends toward the normal. When it enters a faster medium, it bends away from it.

Torque

Torque measures how effectively a force causes rotation about a pivot. It depends on both how large the force is and how far from the pivot it acts, with angle built into the lever effect.

τ = rF sin θ

This is why pushing at the end of a wrench works better than pushing near the center. More lever arm means more turning effect.