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distance between two events in space

is the same in any Euclidean frame, Δs² = Δs'²

Galilean Transformation, x'

x - vt

T

1/f

Einstein's 1st Postulate

All of the laws of physics are the same in every inertial frame

Einstein's 2nd Postulate

The speed of light in a vacuum is the same for all observers

γΔt'

Time Dilation, Δt

γ

1 / √(1 - v²/c²)

Shortest time interval between two events

measured by a clock present at both events, known as proper time interval

L₀/γ

Length Contraction, L

Lorentz Transformation, t'

γ(t - vx/c²)

Lorentz Transformation, t

γ(t' + vx'/c²)

Lorentz Transformation, x'

γ(x - vt)

Lorentz Transformation, x

γ(x' + vt')

Speed, Ux

(U'x + v)/(1 + (vU'x / c²))

Motion at an angle, Uy

U'y / γ(1 + (vU'x / c²))

v₀ √((c-v)/(c+v))

Frequency of pulses, v₁

mc²

Rest Energy, E₀

Mass-energy, E

K + E₀ = γmc²

(γ-1)mc²

Kinetic Energy, K

γmv

Relativistic Momentum, p

p²c²

E² - m²c⁴

Centre of Mass, R

(1/M) ∫[body] r dm

In 1 dimension

dm = ρ dl

In 2 dimensions

dm = ρ da

In 3 dimensions

dm = ρ dV

Conditions for Static Equilibrium

Vector sum of all external forces acting is zero and the sum of all turning moments about an axis through any point is zero

Law of the Lever

Gext = ∑Gi = ∑fi li = 0

Moment Vector, G

Iω' = F x r, where F is the Force acting perpendicular to the axis and r is the distance from the point

Centripetal Force, F

-mω²r = -mv²/r (towards centre of circle)

Angular Moment, Moment of Momentum, L

Iω = r x P = r x mv (perpendicular to r and v (or P))

conservation of angular momentum

just as linear momentum is conserved, so is angular momentum

Angular Impulse, J

r x I = ∫ G dt = ΔL

Moment of Inertia I,

∑mi ri² = ∫[body] r² dm (write dm in terms of dr using density)

Parallel Axis Theorem

The moment of inertia of an axis, AB, parallel to n axis, OP through the centre of mass about which the moment of inertia I₀, is given by:

Perpendicular Axis Theorem

Iz = Ix + Iy, where the lamina lies in the xy plane

½ Iω²

Total Rotational Kinetic Energy