|
s = speed
|
t = time
|
|
Total Distance
|
x
f+xi |
|
One Dimensional Motion
|
|
|
Distance
|
d = s⋅t
|
|
Displacement
|
x
f-xi |
|
Speed
|
(x
f+xi) / (tf+ti`) |
|
Not Constant Velocity
|
|
|
Average Velocity
|
(x
f-xi) / (tf-ti`) |
|
x↑: v+
x↓: v- x→: v=0 |
a+: v↑
a-: v↓ a=0: v→ |
|
Instantaneous Acceleration
|
|
|
(v
f-vi) / (tf-ti`) |
|
|
Constant Acceleration in 1D
|
|
|
V
f = Vi + (a⋅t) |
|
|
Constant Acceleration Final Distance
|
|
|
X
f= 1/2(Vf-Vi) ⋅ t |
|
|
X
f= Xi + (Vi⋅ t) + 1/2(a ⋅ t) |
|
|
a = (V
f-Vi) / t |
t = (V
f-Vi) / a |
|
V
f = Vi⋅ a2 |
|
|
V
f2 = Vi2 + 2⋅a (xf-xi) |
|
|
G
y = -9.8 m/s |
|
|
Angular Frequency
|
w = 2πf
2π/T |
|
Frequency
|
f = 1 / T
|
|
Period
|
T = 1 / f
|
|
Restoring Force
|
F
x = -kx |
|
Simple Harmonic Motion
|
|
|
k = Spring Constant
|
x = displacement
|
|
m = mass
|
|
|
Displacement as function of time
|
x = Acos(wt + Θ)
|
|
Velocity as function of time
|
v = -wAsin(wt + Θ)
|
|
Acceleration as function of time
|
a = -w2Acos(wt + Θ)
|
|
x
max = A [Amplitude] |
-x
max = A [Amplitude] |
|
v
max = wA |
-v
max = wA |
|
a
max = w2A |
-a
max = w2A |
|
Equation for Simple Harmonic Motion
|
a`x = - (k/m) x
|
|
k = restoring force
|
|
|
Angular Frequency for SHM
|
w = √k/m
|
|
Frequency for SHM
|
f = w/2π
|
|
f = 1/2π√k/m
|
|
|
Period for SHM
|
T = 1/f
|
|
T = 2π/w
|
|
|
T = 2π√m/k
|
|
|
Total Mechanical Energy (Constant)
|
E = 1/2mv
x2 + 1/2kx2 |
|
E = 1/2kA2
|
|
|
1km = 1000m
|
1 kg = 1000g
|
|
Dot Product
|
P = Power
|
|
A⋅B = (A
i⋅Bi)+(Aj⋅Bj) |
t = s
|
|
Work = Force ⋅ distance
|
|
|
W = F
x⋅ distance |
|
|
W = F⋅cosΘ⋅distance
|
|
|
K
E: 1/2⋅m⋅v2 |
U = m⋅g⋅h
|
|
W
total = KEf - KEi |
|
|
W
x = F (cosΘ)⋅s || Wy = F (sinΘ)⋅s |
|
|
Constant Speed
|
|
|
Friction (opposite) = cos(180o)
|
|
|
P = F⋅v
|
P = (W/t)
|
|
P
av= ΔW / Δt [Average Power] |
if F→ & s← = - W
|
|
if F↓ & s→ = 0
|
if F→ & s→ = W
|
|
Force Required to
Stretch a spring |
|
|
F
x = k ⋅ x |
|
|
G
E= 6.67⋅10-11 |
Earth Gravity Constant
|
|
R
E= 6.38⋅106 m |
Earth Radius
|
|
M
E= 5.972⋅1024 kg |
Mass of Earth
|
|
g = 9.8 m/s; a
g = 9.8 m/s |
r - R
E = h |
|
F
g = (GE⋅m1⋅m2) / (r2) |
F
g = m ⋅ a |
|
w = m⋅g
|
s = r - R
E cosΘ |
|
Gravitation and Spherically
Symmetric Bodies |
F
g = (GE⋅mE⋅m) / (r2) |
|
Weight of the body at Earth's Surface
|
w = F
g = (GE⋅mE⋅m) / (RE2) |
|
Acceleration due to Gravity
|
g = (G
E⋅mE) / (RE2) |
|
Velocity of Earth
|
V
E= 4/3πRE2 = 1.08⋅1021 m3 |
|
Gravitational Potential Energy
|
|
|
U = -(G
E⋅mE⋅m) / (r) |
|
|
WorkDone by Gravity
|
W
grav = m⋅g(r1-r2) |
|
W
grav = GmE⋅m ⋅ (r1-r2) / (r1⋅r2) |
|
|
W
grav= GmE⋅m ⋅ (r1-r2) / (RE2) |
[if the body stays close to Earth]
|
|
Speed of the Satellite
|
v = √(G⋅m
E / r) |
|
Period of Circular Orbit
|
T = (2πr / v)
|
|
T = 2πr3/2/√G⋅m
E |
T = 2πr √(r / G⋅m
E) |
|
Point Mass Outside
a Spherical Shell |
U
i= - Gm⋅mi / s |
|
Apparent weight
; Earth's Rotation |
|
|
w
0 = true weight of object |
F = force exerted by spring scale
|
|
F + w
0 = net force on object |
w = apparent weight = opposite of F
|
|
centripetal acceleration`
|
w
0- F = (mv2 / RE) |
|
w = w
0 - (mv2 / RE) |
|
|
freefall acceleration
|
g = g
0 - (v2/ RE) |
|
Black Holes
|
|
|
P = Density
|
P = M / v
|
|
v = 4/3πR3
|
c = speed of light in the vaccum
|
|
Schwardzschild Radius
|
R
s = 2GM / c2 |
|
c = √2GM / R
S |
|
|
Y-axis
E = Mechanical Energy |
|
|
W
grav = F ⋅ s = w(y1-y2) |
|
|
W
grav=(m⋅g⋅y1)-(m⋅g⋅y1) |
|
|
W
grav=Ugrav,1 - Ugrav,2 |
|
|
W
grav = -Δ Ugrav |
|
|
Conservation
of Mechanical Energy |
|
|
K
f-Ki = Ugrav,1 - Ugrav,2 |
|
|
K
i+Ugrav,1=Kf+Ugrav,2 |
|
|
E = K + U
grav = constant (if gravity does work) |
|
|
When other forces
than Gravity do work |
|
|
W
other + Wgrav = Kf - Ki |
|
|
Elastic Potential Energy
|
|
|
U
el = 1/2kx2 |
|
|
Work Done a Spring
|
|
|
W = 1/2kx
22 - 1/2kx12 |
|
|
If Elastic does work,
total mechanical energy is stored |
|
|
K
i+Uel,1=Kf+Uel,2 |
|
|
Situations with Both Gravitational
and Elastic Potential Energy |
|
|
K
1+U1+Wother=K2+U2 |
|
|
The work done by all forces other than
the gravitational force or elastic force equals the change in total mechanical energy E = K + U of the system |
|
|
The Law of Conservation
of Energy |
|
|
ΔU
int = -Wother ΔU int = internal energy |
|
|
Force and Potential Energy
|
|
|
F
x(x) = - dU(x) / dx |
|
The Simple Pendelum (TSP)
|
L = pendulum length
|
|
Angular Frequency TSP
|
w = √k/m
|
|
w = √mg / L /m
|
|
|
w = √g/L
|
|
|
Frequency TSP
|
f = w/2π
|
|
f = 1/2π √g/L
|
|
|
Period TSP
|
T = 2π/w
|
|
T = 1/f
|
|
|
T = 2π√L/g
|
|
|
The Physical Pendulum (TPP)
|
|
|
L = angular momentum
|
L = mvr
|
|
w = Angular Velocity
|
w = ΔΘ / Δt
|
|
(I)nertia = L / w
|
|
|
Angular Frequency TPP
|
w = √mgd / I
|
|
Period TPP
|
T = 2π √ I / mgd
|
|
Damped Oscillation
|
|
|
b = Damping Constant
|
|
|
Displace of Damped
|
x = Ae-b(2m)t cost (wt + Θ)
|
|
Angular Frequency of Damped
|
w' = √ (k/m) - (b2 / 4m2)
|
|
Force Oscillations and Resonance
|
|
|
F
max = Maximum Driving Force |
k = constant restoring force
|
|
w
d = Driving Angular Frequency |
|
|
A = F
max / √(k-mwd2)2 + b2wd2 |
|
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