ap physics m formulas

time independent kinematics equation

v_f² = v_o² + 2ax

net force (Newton’s second law)

∑F = ma

force in terms of momentum (Newton’s 2nd law as a derivative)

F = dp/dt

force in terms of potential energy

F = -dU/dx

impulse

J = ∫F dt

definition of momentum

p = mV

impulse - momentum theroem

J = ∆p

force of friction

F_ƒ ≤ µF_N

work done by a constant force (dot product)

W = F * d

work done by a variable force (integral)

W = ∫F * ds

kinetic energy (linear)

E_K = (1/2)mv²

work - energy theorem

W_net = ∆E_k

power (as a rate of change)

P = dW/dt

power (dot product)

P = F * v

centripetal acceleration

a_c = v²/r = ω²r

torque (cross product)

τ = r x F

Newton’s second law for rotation (torque and angular acceleration)

∑τ = Iα

moment of inertia of a collection of particles (no integral)

I = ∑m_ir_i²

parallel axis theorem

I_‖ = I_com + mh²

rotational inertia of a rod about an axis through its center

I_rod = (mℓ²)/12

/ ℓ is not a vector

angular momentum of a moving particle (cross product)

ℓ = r x p

/ ℓ is a vector

angular momentum of a rigid rotation body (rotational inertia)

L = Iω

position of a center of mass for a collection of particles (sigma notation)

r_com = (∑m_ir_i)/M

conversion between linear and angular velocity (no slip)

ω x r = V → V = rω

rotational kinetic energy

E_k = (1/2)Iω²

force of a spring (Hooke’s law)

F = -kx

potential energy of a spring

U_spring = (1/2)kx²

period of a spring mass system

T = 2π √(m/k)

angular frequency of a general pendulum

ω = √(MgD/t)

period of a simple pendulum

T = 2π √(ℓ/g)

/ ℓ is not a vector

relationships between period, frequency, and angular frequency

1/T = ƒ = ω/(2π)

Newton’s law of gravitation

F_G = (Gm₁m₂)/r²

gravitational potential energy

U_G = (-Gm₁m₂)/r

total mechanical energy of an object in circular orbit

U_total = (-Gm₁m₂)/2r

Kepler’s 3rd law

(τ²)/(r³) = (4π²)/(GM_s)

/ note: r is average of r_min and r_max

escape velocity

v_escape = √((2GM_e)/R_e) = √(2R_eg)