AP Physics Mechanics Equations to Memorize Flashcards by curt bixel

Integral of a differential

∫ dx = x + C

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Derivative chain rule

d/dx (u) = (du / dv)(dv / dx)

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Derivative quotient rule

d/dx (u / v) = (1/v)(du/dx) – (u/v2)(dv/dx)

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Spring force

F_sp = -ks

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Rule for the angle of the cross product

Rotate counterclockwise from the first vector to the second vector

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Conservation of angular momentum

I₁ω₁ = I₂ω₂

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Centripetal acceleration based upon ω

a_c = ω²R

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Relationship between the force and the change in energy per unit distance.

F = -dU/ds

This is a combination W = f x d and W = -ΔU

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Rotational inertia of point masses

I = Σ(mr²)

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Torque (2)

τ = F∙ r_┴

( torque = force times "lever arm" )

(The lever arm is just the shortest distance between the axis of rotation and the path the force is acting along.)

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Cartesian to polar coordinates (2)

v = √(v_x² + v_y²)

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Rotational work

W_rot = (τ)(Δθ)

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Integral of an exponential term

∫ (e^u)dx = (1/u’)(e^u) + C

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Slope of a velocity/time graph

Acceleration

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Period of a simple pendulum

T = 2π√(L/g)

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Centripetal force

F_c = ma_c

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Integrating with a constant

∫ k f(x)dx = k ∫ f(x)dx

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Kinetic friction

F_kf = ± μ_kf∙ F_N

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Period of a physical pendulum

T = 2π√(I/mgd)

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Gravitational potential energy on a planet

U_g = mgh

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Acceleration due to gravity

g = G∙m_p/r_p²

(This is pretty much just the Law of Universal Gravitation with the mass of the planet and the radius of the planet plugged in.)

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Gravitational potential energy in space

U_g = (-G∙m₁∙m₂)/ r

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Universal force of gravity

F_g = (G∙m₁∙m₂)/ r²

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Acceleration

a = dv / dt

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Slope of a potential energy / position graph

Negative of force

Displacement under constant acceleration

Δs = (v_i)(Δt) + ½(a)(Δt²)

Power input by a force

P = W / Δt

Force down an incline

F_ll = F_g ∙ sinθ

Area under an acceleration/time function

Change in velocity

Sum rule for integration

∫ (u + v)dx = ∫ (u)dx + ∫ (v)dx + C

Derivative of sin

d/dx (sin x) = cos x

Area under a force/position function

Work

Escape velocity

v = √2Gm / R

Constant acceleration

a = Δv / Δt

Derivative of cos

d/dx (cos x) = - sin x

Kinetic energy

K = ½ mv²

v_f² equation

v_f² = v_i² + 2(a)(Δs)

Potential energy in a spring

U_sp= ½ ks²

Displacement

Δs = s_f – s_i

Integral of sin

∫ (sin x)dx = -cos x + C

Horizontal vector component

v_x = v(cos θ)

Formula for finding initial vertical velocity of a projectile given its initial velocity and angle at which it is fired.

v_y = v(sin θ)

Force in a gravitational field

F_g = -mg

Power rule for integration

∫ f(xⁿ)dx = (xⁿ⁺¹/n+1) + C

Conservation of angular momentum

ΔL=0

if and only if

Στ_ext = 0

Rule for the angle of the dot product

Rotate counterclockwise from the first vector to the second vector

Work done by a variable force

W = ∫(F)(ds)

Newton’s second law for rotation

Στ = I∙α

( α is alpha, or angular acceleration )

Work – potential energy relationship

-W = ΔU

Momentum

p = mv

Center of mass

r_cm = (m₁)(r₁) + (m₂)(r₂) … / Σm

Rotational kinetic energy

K_rot = ½∙I∙ω²

Relative motion

v_a,c = v_a,b + v_b,c

Integrating 1/x

∫ (1/x)dx = ln|x| + C

Angular momentum (2)

L = rmv

also

L = Iω

Impulse equation

J = Δp

( Impulse = change in momentum )

Satellite velocity

v_sat = √(G∙m_centeral / r_orbit)

Instantaneous power

P = F∙v

Slope of a momentum/time graph

Force

Derivative sum rule

d/dx (u + v) = du/dx + dv/dx

Relationship between period and frequency

T = 1/f

Area under a force/time function

Impulse

Static friction

F_sf ≤ ± μ_sf ∙ F_N

Integral of cos

∫ (cos x)dx = sin x + C

Work (2)

W = (F_ll)(Δs)

Impulse for a constant force

J = (ΣF)(Δt)

(Impulse = force x time)

Work – energy theorem

ΣW = ΔK

( make sure you remember this one )

Velocity

v = ds / dt

Newton’s third law

F_ab = -F_ba

Gravitational field lines

Vectors point how a test mass would accelerate

Parallel axis theorem

I = I_cm + md²

( If you know the moment of interia of an object rotating around its center of mass I_cm but the object is instead rotating around an axis that is distance "d" from the center of mass, the new rotational intertia can be found with this equation )

Average velocity

v_avg = Δs / Δt

Area under a velocity/time function

Change in position

Conservation of energy

ΣU + ΣK + ΣEth = constant for a closed system

( this is a simplification, as it does not include chemical potential energy, electrical potential energy, etc. )

Period of a spring oscillator

T = 2π√(m/k)

Centripetal acceleration based upon v

a_c = v² / r

Speed

S = distance / time

Impulse for a variable force

J = ∫(F)(dt)

Newton’s second law

ΣF = ma

(don't forget how many times you could get 1 point on a free response simply by writing this down)

Angular frequency (2)

ω = 2πf

Slope of a position/time graph

Velocity

Average velocity when acceleration is constant

v_avg = (v_i + v_f) / 2

Force perpendicular to an incline

F_N = F_g ∙ cosθ

Conservation of momentum

Σp_i = Σp_f if ΣF_ext = 0

( With no external forces, the momentum of a system will be conserved )

( This is true for both linear and angular momentum )

Power (general)

P = ΔE / Δt

Rotational instantaneous power

P = τ·ω

Derivative product rule

d/dx (uv) = v(du/dx) + u(dv/dx)

Rotational inertia of radially symmetric objects

I = kMR²

^{( k = the number of these objects )}

Equations for rolling (3)

ds = r∙dθ