OCR MEI Mechanics Major

centre of mass of a solid hemisphere

3/8 r from base

centre of mass of a solid cone

1/4 h

volume of solid cone

1/3 πr²h

centre of mass of a hollow hemisphere

1/2 r

centre of mass of a hollow cone

1/3 h

volume of a hollow cone

πrl

Ax̄, where A = ∫y dx

∫xy dx

Aȳ, where A = ∫y dx

∫y²/2 dx

Ax̄, where A = ∫x dy

∫x²/2 dy

Aȳ, where A = ∫x dy

∫xy dy

Vȳ, where V = π∫y² dx

∫y²x dx

Vx̄, where V = π∫x² dy

∫x²y dy

Time period of a simple pendulum

T = 2π√(l/g)

Time period of a mass-spring system

T = 2π√(m/k)

F = μR

maximum possible friction

friction during active motion

limiting equilibrium

F ≤ μR

general case for static friction

when an object is at rest and not necessarily about to move

relationship for sliding

μ = tanθ

relationship for toppling

tanθ = width/length

equations for rotational dynamics

x = Acosωt

v = ± 𝜔 √(A² + x²)

vₘₐₓ = ωA

aₘₐₓ = ω²A

a = - ω²x