Phys 109
Final Examination Equations
vx = vx0 + ax t
vy = vy0 + ay t
2
+ 2ax (x − x0 )
vx2 = v0x
2
+ 2ay (y − y0 )
vy2 = v0y
x = x0 + vx0 t + 12 ax t2
y = y0 + vy0 t + 12 ay t2
x = x0 + 12 (vx0 + vx )t
y = y0 + 12 (vy0 + vy )t
Gravitational acceleration near the surface of the Earth: g = 9.8 m/s2 .
P
F~net = ni=1 F~i = m~a
I~ = F~ ∆t = ∆~p
~|
|f~s(max) | = µs |N
K = 21 mv 2
~|
|f~k | = µk |N
|F~Hooke | = −kx
~ cos φ = ∆K
W = F~ · d~ = |F~ ||d|
|V~ |2 = V 2 = Vx2 + Vy2
Emech = K + U
Wcons = −∆U
Wnc = ∆U + ∆K
UHooke = 12 kx2
UG = mgh
θ = θ0 + ω0 t + 21 αt2
ω = ω0 + αt
vtan = ωr
atan = αr
Krot = 21 Iω 2
Wrot = τ θ
P̄ =
W
∆t
ω 2 = ω02 + 2α(θ − θ0 )
acent =
I=
Pn
i=1
v2
r
mi ri2
τ = |F~ ||~r| sin(φ) = Iα
L = |~r||~p| sin(φ) = Iω
1
f
=
1
di
= (n − 1)
+
1
do
1
R1
−
f=
1
R2
R
2
n1 sin(θ1 ) = n2 sin(θ2 )
m=
hi
h0
Pn
xCM =
1
M
|~anet | =
q
i=1
= − ddoi
m i xi
a2cent + a2tan
Prot = τ ω
θ = θ0 + 21 (ω + ω0 )t
1
f
= F|| v̄
2
3
Sign convention for all lenses:
→ do is + if the object is in front of the lens.
→ do is − if the object is in back of the lens.
→ di is + if the image is in back of the lens. (Real)
→ di is − if the image is in front of the lens. (Virtual)
→ R1 and R2 are − if the centre of curvature is in front of the lens.
→ R1 and R2 are + if the centre of curvature is in back of the lens.
→ m is + if the image is right side up.
→ m is − if the image is upside down.
Sign convention for all mirrors:
→ do is + if the object is in front of the mirror.
→ do is − if the object is in back of the mirror.
→ di is + if the image is in front of the mirror. (Real)
→ di is − if the image is in back of the mirror. (Virtual)
→ f and R are + if the centre of curvature is in front of the mirror. (Concave)
→ f and R are − if the centre of curvature is in back of the mirror. (Convex)
→ m is + if the image is right side up.
→ m is − if the image is upside down.
Useful Math
2
ax + bx + c = 0 −→ x =
Vx = V cos(θ)
θ = arcsin
θ = arctan
Vy
V
= sin−1
Vy
Vx
= tan−1
Vx
V
√
b2 − 4ac
2a
Vy = V sin(θ)
cos(θ) =
−b ±
Vy
V
Vy
Vx
θ = arccos
Vx
V
= cos−1
sin(θ) =
Vy
V
tan(θ) =
Vy
Vx
Vx
V
sin(2x) = 2 sin(x) cos(x)
4
cos(2x) = cos2 (x) − sin2 (x) = 2 cos2 (x) − 1 = 1 − 2 sin2 (x)
1 + cos(2x)
1 − cos(2x)
cos2 (x) =
sin2 (x) =
2
2
sin(β)
sin(γ)
sin(α)
=
=
C 2 = A2 + B 2 − 2AB cos(γ)
A
B
C