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I. CONSTANTS

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• Acceleration of earth gravitational field: g = 9.8m/s .

II. GENERAL MATH

• Quadratic equation: ax2 + bx + c = 0 is solved by

√
−b ± b2 − 4ac
x= .
2a
• Components of vector with magnitude A and θ the angle with x-axis:
Ax = A cos θ , Ay = A sin θ .

~ = (Ax , Ay ):
• Magnitude of vector A
q
~ =
A = |A| A2x + A2y .

• Angle from x-axis:

Ay
tan θ = .
Ax
• Scalar vector product (θ angle between vectors)
~a · ~b = abcosθ = ax bx + ay by + az bz

• Cross vector product (φ angle between vectors)

~c = ~a × ~b ~c ⊥ ~a, ~b c = ba sin φ

• Integrals
xn+1
Z
xn dx =
n+1
• radian: 1 rad = 180◦ /π

III. MOTION IN A STRAIGHT LINE

• Velocity
∆x dx
vav,x = vx = .
∆t dt
• Acceleration
∆vx dvx d2 x
aav,x = ax = = 2.
∆t dt dt
• Constant acceleration:
General Free fall (axis pointing upwards)
vx = v0x + ax t vy = v0y − gt
ax t2 gt2
x = x0 + v0x t + y = y0 + v0y t −
2 2
vx2 = v0x
2
+ 2ax (x − x0 ) vy2 = v0y
2
− 2g(y − y0 )
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IV. MOTION IN 2D, 3D

• 3D position vector: ~r = (x, y, z).

• Velocity
∆~r d~r
~vav = ~v =
∆t dt

• Acceleration
∆~v d~v d2~r
~aav = ~ax = = 2
∆t dt dt

• Projectile motion (release velocity ~v0 , angle of release with horizontal α)

gt2 gt2
x = x0 + v0x t = x0 + v0 cos αt y = y0 + v0y t − = y0 + v0 sin αt −
2 2
vx = v0x = v0 cos α vy = v0y − gt = v0 sin α − gt

• Projectile trajectory (release in origin):

gx2
y = tan α x −
2v02 cos2 α

• Circular motion with constant speed, radius R and revolution time T :

v2 2πR
arad = v=
R T

V. FORCES – NEWTON’S LAWS

F~tot =
P~
• Newton’s first law (object in equilibrium) F =0

F~tot = F~ = m~a
P
• Newton’s second law

• Newton’s third law F~1 on 2 = −F~2 on 1

• Kinetic friction force fk = µk FN
• Static friction force fs ≤ µs FN

VI. WORK AND ENERGY

• Work

Constant Force - Straight Line W = F~ · ~s

Z
Varying Force - Curve W = F~ · d~x

• Pulling force on a spring F = kx

• Gravitational work Wg = mg(y1 − y2 )
• Work done by a spring on attached object Wel = 21 k(x21 − x22 )

• Kinetic energy K = 12 mv 2
• Work-kinetic energy theorem Wtot = K2 − K1
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• Power P = dW
dt = F~ · ~v
• Conservation of mechanical energy (“other” are non-conservative forces) Wother + K1 + U1 = K2 + U2
• Gravitational potential energy Ug = mgy
• Spring elastic potential energy Usp = 21 kx2

• Force from potential energy [1D] Fx = − dUx(x)

VII. MOMENTUM AND IMPULS

• Momentum p~ = m~v
• Impuls

Constant force J~ = F~tot ∆t

Z t2
Varying force J~ = F~tot dt = F~tot,av (t2 − t1 )
t1

• Momentum-impuls theorem J~ = ∆~
p

• Conservation of total momentum P~ (2 particles) mA~vA1 + mB ~vB1 = mA~vA2 + mB ~vB2

• Elastic collision in 1D only vA1 − vB1 = −(vA2 − vB2 )
• Center of Mass (total mass M )
P
mi~ri X
~rCM = i P~ = M~vCM F~ext = M~aCM
M

VIII. ANGULAR MOTION

dθ vT
• Angular velocity ω= dt = R
dω aT
• Angular acceleration α= dt = R

• Radial acceleration aR = Rω 2
• Motion with constant angular acceleration:

t2
θ = θ0 + ω0 t + α
2
ω = ω0 + αt
ω 2 = ω02 + 2α(θ − θ0 )

• Kinetic energy of rotation Krot = 21 Iω 2

I = i mi ri2
P
• Moment of inertia
• Parallel axis theorem (d distance between axes) IP = Icm + M d2
• Useful moments of inertia
1 2
– Slender rod with length L (axis through center) I= 12 M L
– Solid cylinder with radius R (axis through central point along cylinder) I = 12 M R2
2
– Hoop with radius R (axis through central point perpendicular to hoop) I = MR
– Solid sphere with radius R (any axis through central point) I = 25 M R2
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IX. ROTATIONAL DYNAMICS

• Torque ~τ = ~r × F~
• Magnitude torque (lever arm l) τ = F r sin φ = F l = Ftan r
• Newton’s second law for rotational motion τ = Iα
• Translation and rotation K = 12 M vCM
2
+ 12 ICM ω 2
• Rolling without slipping vCM = Rω
• Work done by constant torque W = τz ∆θ
• Work - Kinetic energy theorem W = 12 Iω22 − 12 Iω12
• Power P = τz ωz
• Angular momentum ~ = ~r × p~ = I~
L ω
~
dL
• Instantaneous torque ~τ = dt

• Conservation of angular momentum (no external net torque) L1 = L2