PH2100 Formula Sheet - Physics

PH2100 Formula Sheet Knight
Concepts of Motion
distance traveled
average speed = time interval spent traveling
∆ r = rf
−ri
∆r
vavg
= ∆ t
∆v
aavg
= ∆ t
Kinematics: The Mathematics of Motion
For Motion in a Straight Line:
ds
vs
= = slope of position-time graph
dt
dvs
as
= = slope of velocity-time graph
dt
tf
⎧area under velocity curve
sf = si + ∫ vsdt = si
+⎨
ti
⎩ from ti
to tf
tf
⎧area under acceleration curve
vfs = vi s
+ ∫ asdt = vi
s
+ ⎨
ti
⎩from
ti
to tf
Uniform Motion:
s = s + v ∆t
f
i
Uniformly Accelerated Motion:
v = v + a ∆t
fs is s
1
f i is
2
v = v + 2a ∆s
2 2
fs is s
s
s
( )
s = s + v ∆ t+ a ∆t
Motion on an Inclined Plane : a =± gsinθ
Vectors and Coordinate Systems
2
s
Force and Motion
1
a = Fnet
m
where F = F + F + F + ...
net 1 2 3
Dynamics I: Motion Along a Line
F = F = ma
( µ
sn, direction as necessary to prevent motion)
( µ
kn, direction opposite the motion)
( µ n, direction opposite the motion)
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net
∑
i
i
A Model for Friction :
Static : fs
≤
Kinetic : fk
=
Rolling : fr
=
r
1 2
Drag : D ≈
4
Av , direction opposite the motion
( )
Dynamics II: Motion in a Plane
r = xiˆ+
yj ˆ
dr dx
ˆ
dy
v = = i + ˆj = v ˆ ˆ
xi + vy
j
dt dt dt
dv dv dv
x y
a = = iˆ+ ˆj = a ˆ ˆ
xi + ay
j
dt dt dt
F = F = ma
( net )
( )
net
x
F = F = ma
y
∑
∑
Constant Acceleration :
( ) ( )
1
2
1
2
f
=
i
+
ix∆ +
2 x
∆
f
=
i
+
iy∆ +
2 y
∆
v = v + a ∆t
fx ix x
x
y
x
y
x x v t a t y y v t a t
v = v + a ∆t
fy iy y
y
Projectile Motion :
1
2
A
xf = xi + vi x∆ t yf = yi + viy∆t− g( ∆t
2 )
Ay
= Ay
ĵ
vf x
= vix = constant vf y
= vi
y
−g∆t
θ 2 2
A ˆ
vf y
= viy
−2g( yf
− y )
x
= Axi
i
x
y y′
A= A ˆ ˆ
x
+ Ay = Axi + Ay
j
Relative Motion:
In the figure above:
rPS = rPS '
+ vS ' St
• P
Ax
= Acos
θ Ay
= Asinθ
vPS = vPS '
+ vS ' S
S
A
2 2 −1⎛
y ⎞
′
x′
A= Ax
+ Ay
θ = tan
⎜ ⎟
⎝ A
aPS
= aPS
'
S
x ⎠
x

Dynamics III: Motion in a Circle
s
θ =
r
∆θ
average angular velocity = ∆ t
dθ
ω =
dt
v = ωr
t
Uniform Circular Motion :
2πr
2 π rad
v = ω =
T T
θ = θ + ω ∆t
f
i
2
v 2
ar
= = ω r
r
Nonuniform Circular Motion (constant a ):
dv
at
=
dt
at
2
θf = θi + ωi∆ t+ ( ∆t)
2r
at
ωf
= ωi
+ ∆t
r
2
mv
2
Fn
et
=
r ∑ Fr
= mar
= = mω
r
r
⎧0 uniform motion
Fnet
= F
t ∑ t
=⎨
⎩ mat
nonuniform motion
( )
( )
( F )
net
z
∑
= F = 0
z
t
Energy
K =
U
g
mech
1
2
2
f f i i
1
( Fsp ) =−k∆ s Us
=
2
k( ∆s)
s
mv
= mgy
E = K + U
K + U = K + U
(conservation of mechanical energy)
Perfectly Elastic 1-D Collisions ( m initially at rest):
m − m 2m
+ +
1
1 2 1
( v ) = ( v ) ( v ) = ( v )
Work
fx 1 ix 1 fx 2
ix
m1 m2 m1 m2
sf
⎧
⎪ Fds
s
=
si
area under the force-position curve
W =
∫
⎨ ⎪
⎩F⋅∆ r = F∆rcos θ if F is a constant force
∆ K = W = W + W + W (work-kinetic energy theorem)
net c diss ext
f i c
th micro micro
th
diss
( i f)
∆ U = U − U =−W
→
dU
Fs
=−
ds
E = K + U
∆ E
E
sys
= −W
= K + U + E
th
Kf + Uf +∆ Eth = Ki + Ui + Wext
dE
dW
= = = ⋅ =
dt
dt
sys
P P F v Fv
2
2
cosθ
Newton’s Third Law
F
A on B
=−F
B on A
Impulse and Momentum
p = mv
f
tf
⎧
⎪∫
Fx
t dt =
ti
( ) area under force curve
J
x
= ⎨
⎪⎩ Favg∆t
∆ px
= Jx
(impulse-momentum theorem)
P = p1+ p2 + p3+
...
P = P (conservation of momentum)
i
Newton’s Theory of Gravity
GMm
FM on m
= Fm on M
=
2
r
GM
gsurface =
2
R
GM ⎛ 4π
⎞
Circular Orbit: v= T =⎜ ⎟r
r ⎝GM
⎠
Physical Constants
g = 9.80 m/s
earth
earth
2
G = 6.67 × 10 N ⋅m / kg
M
R
= ×
−11 2 2
24
5.98 10 kg
= ×
6
6.37 10 m
2
2 3
Need additional help? Then visit the Physics Learning Center located in 228 Fisher. Walk-in hours are Sunday
7:00 – 9:00 p.m. and Monday through Thursday 3:00 - 9:00 p.m.