Physics Olympiad

Important formulas for the physics olympiad.

Vis-viva equation

v velocity of orbiting object
r distance between orbiting bodies
a semi major axis of the orbit

Period of Orbit

T orbital period
a semi major axis

Capstan Equation

T₁​ smaller tension
T₂​ larger tension
μ coefficient of friction
θ wrap angle in radians

Euler’s Identity

Repeated Root Solution 2^nd Order DE

r repeated root

note: use this if the discriminant of the characteristic equation (ar2+br+c=0) is 0. Plug in the repeated root and use initial conditions to solve.

Final Velocities for Elastic Collision

u₁ object one’s initial velocity
u₂ object two’s initial velocity

note: v₁ had the velocity initially

Range Formula

Maximum Height

Projectile Time

Translating & Rotating Angular Momentum

MRxV angular momentum from the orbit around origin
L angular momentum of the object around its own COM

note: MRxV accounts for the whole objects movement around the origin, L accounts for the object spinning about itself

Translating & Rotating Angular KE

½ MV² angular momentum from the orbit around origin
½ Iw² angular momentum of the object around its own COM

note: MRxV accounts for the whole objects movement around the origin, L accounts for the object spinning about itself

Reduced Mass

note: reduced mass is used for treating multiple free masses on a spring as one

Double Angle Identities

Semi Major Axis

r_p periapsis
r_a apoapsis
a semi-major axis

Simple Harmonic Motion Periods

Centroid of Equilateral Triangle

2/3 the height from a vertex

Triangular Spring Setup to Delta

k’=3k

k’ initial spring constant

k final spring constant

the triangle turns into a Y shape! Easier to work with

Acceleration of Rolling w/o Slipping on an Incline

k = coefficient for rotational inertia of object

Infinite Geometric Series

a first term
r common ratio (the one increasing in power)

Critical Slip Angle for Object on Incline

COM of Polygon

Critical Tipping Angle of Objects on Incline

a half-width

h height of com

Error Propagation Sum

Error Propagation Product

Error Propagation Power

Atwood Machine Acceleration and Tension

String Property of Ellipses

Period of Physical Pendulum

d distance from pivot to COM

Coriolis Force