Common Prefixes
Value Name Symbol
10−9
nano n
−6
10
micro µ
10−3
milli
m
−2
10
centi c
−1
10
deci
d
103
kilo
k
6
10
mega M
109
giga
G
Common
Length
1 in = 2.540 cm
1 m = 39.37 in
1 m = 3.281 f t
1 f t = 0.3048 m
1 km = 0.6214 mile
1 mile = 1.609 km
1 mile = 1609 m
Conversions
Force
1 N = 0.2248 lb
Energy
1 cal = 4.184 J
1 kW h = 3.60 × 106 J
Power
1 hp = 746 W
On Earth: A mass of 1 kg equals a weight of 2.20 pounds
~ = (A, θ) : magnitude and angle (CCW) with +X axis.
Polar: A
~ = (Ax , Ay , Az ) or A
~ = Ax î + Ay ĵ + Az k̂
Cartesian: A
~·B
~ = |A||B| cos φ (angle between the vectors).
Scalar Product: A
~ =A
~ × B.
~ Note A
~×B
~ = −B
~ × A.
~
Vector Product: C
Cx = Ay Bz −Az By
Cy = Az Bx −Ax Bz
Cz = Ax By −Ay Bx
Velocity average: vavg = ∆x/∆t
Acceleration average: aavg = ∆v/∆t
instantaneous: v = dx/dt.
instantaneous: a = dv/dt.
Straight-line (i.e. 1-D) motion with constant acceleration.
v = vo + at
vavg = vo + 21 at
vavg = ∆x/∆t
1 2
x = xo + vo t + 2 at
x = xo + vavg t
vavg = 12 (vo + v)
v 2 = vo2 + 2a(x − x0 )
Freely-falling bodies: ~a = ~g , where g = 9.80 m/s2 .
Generic Equations and definitions (even when ~a varies)
~v = d~r/dt
~vavg = ∆~r/∆t
~aavg = ∆~v /∆t
~r = xî + y ĵ + z k̂
~r = ~ro + ~vavg t
vx = dx/dt
vy = dy/dt
vz = dz/dt
ax = dvx /dt
ay = dvy /dt
az = dvz /dt
Vector Equations of Motion (constant ~a )
vx = vox + ax t
vy = voy + ay t
vz = voz + az t
1
1
2
2
x = xo + vox t + 2 ax t y = yo + voy t + 2 ay t z = zo + voz t + 12 az t2
~v = ~vo + ~at
~r = ~ro + ~vo t + 21 ~at2
v 2 = vo2 + 2ax ∆x + 2ay ∆y + 2az ∆z
2
2
2
+ 2ax ∆x
+ 2ay ∆y
+ 2az ∆z
vx2 = vox
vy2 = voy
vz2 = voz
Projectile Motion: ~a = ~g with g = |~g | = +9.80 m/s2 . If +Y points upward (ax = 0, ay = −g) then:
x = xo + vox t
vx = vox
y = yo + voy t − 21 gt2
vy = voy − gt
If launched at origin with ~vo = (vo , θ) : vox = vo cos θ
voy = vo sin θ
g
1 2
2
y = (tan θ)x − ( 2v2 cos
x = (vo cos θ)t
y = (vo sin θ)t − 2 gt
2 θ )x
o
2
Apogee: h = (vo2 sin θ)/(2g) above the launch point. tA = (vo sin θ)/(g)
Rmax = vo2 /g at θ = 45o
If yf inal = yinitial : Range R = (vo2 sin 2θ)/(g)
Uniform Circular Motion:
Radial acceleration: ar = v 2 /r
Period: T = 2πr/v
or
Total flight time: t = 2tA
v = 2πr/T
Newton’s First Law: ΣF~ = 0 implies ~a = 0 (and vice versa).
Newton’s Second Law: ΣF~i = m~a. (I.e., ~a = (ΣF~i )/m)
Newton’s Third Law: F~1 on 2 = −F~2 on 1 (interaction)
P
P
Equilibrium: At every point Fx = 0 and Fy = 0 separately. (Motion ok, just no acceleration.)
Friction: fk = µk n
fs ≤ fs,max = µs n
Direction opposes motion.
Work: W = F~ · d~
Kinetic Energy: K = 21 mv 2
Average Power: Pavg = ∆W/∆t
Work and Power
R
1-D Varying Force: W = if Fx dx
P
Work - Kinetic Energy Theorem: K = Ko + Wi
Power: P = dW/dt = F~ · ~v
Potential Energy : Ug = mgh
Us = 12 kd2 (careful with h and d)
Mechanical Energy E = K + Ug + Us
(K = Kcm + Krot if object rotates)
Conservation of Energy (K + Ug + Us )2 = (K + Ug + Us )1 + ΣWother
Linear Momentum of a moving object: p~ = m~v
P
Generalized Newton’s Law: F~ = d~p/dt
Impulse: J~ = ∆~p = p~2 − Rp~1
Constant force: J~ = F~ ∆t
t
f
~
Time-varying force: J~ = ti F~ dt = F~avg ∆t
|
F~avg = J/∆t
or F~avg = ∆~p/∆t.
P
P
P
P
Collisions : Total momentum is conserved: p~i = p~f ie mi~vi = mf ~vf
Inelastic Collision : some mechanical energy is lost (most common case)
Perfectly inelastic : objects stick together, move off as single object. Max loss of energy.
Elastic collisions: the momentum and the total kinetic energy are both conserved (rare).
Trig: sin (2θ) = 2 sin θ cos θ
Quadratic formula: if Ax2 + Bx + C = 0 then x =
√
−B± B 2 −4AC
2A
Analogies between Rotational and Linear Motion
Rotational Motion About a Fixed Axis
Linear Motion
Angle θ (radians)
Position x (meters)
Angular speed ω = dθ/dt (rad/sec)
Linear speed v = dx/dt (m/s)
Avg Angular speed ωavg = ∆θ/∆t (rad/sec) Avg Linear speed vavg = ∆x/∆t (m/s)
Angular acceleration α = dω/dt (rad/sec2 ) Linear acceleration a = dv/dt (m/s2 )
Rotational kinetic energy Krot = 12 Iω 2
Kinetic energy Ktrans = 12 mv 2
Rotational work (constant τ ) W = τ θ
W = F~ · d~
Rotational power (constant τ ) P = τ ω
P = F~ · ~v
For CONSTANT α
For CONSTANT a
ω = ωo + αt
v = vo + at
1
2
θ = θo + ωo t + 2 αt
x = xo + vo t + 21 at2
θ = θo + 12 (ωo + ω)t
x = xo + 12 (vo + v)t
ω 2 = ωo2 + 2α(θ − θo )
v 2 = vo2 + 2a(x − xo )
Total kinetic energy: K = Ktrans + Krot
Conversion: 360o = 2π radians
Rigid body rotation: s = rθ
vt = rω
at = rα.
2
2
ar = v /r = ω r
ω = 2π/T
T = 2π/ω
f = 1/T
Center of mass: Xcm = (Σmi xi )/(Σmi )
Ycm = (Σmi yi )/(Σmi )
Moment of Inertia Collection of particles: I =
Parallel axis: I = Icm + M d2
Torque: force acting at a point P on an
object that (may) rotate about a point O
~τ = ~r × F~
Σ~τ = I~
α, with I computed about the rotation axis through O. If all forces are in
the XY plane, rotation will be about the
Z axis: Στ = Iα (all computed about Z).
Note: Fg acts at CM of object.
Static Equilibrium:
• ΣF~ = 0 for every object
• Σ~τ = 0 about any axis
P
mi ri2
Zcm = (Σmi zi )/(Σmi )
Solid object: I = r2 dm
R