CC1529
Semester 1, 2015
Page 1 of 3
PHYS1001 Physics 1 (Regular) Formula Sheet
Vectors
Kinematics
~ = Ax î + Ay ĵ + Az k̂
A
q
~ A = A
= A2x + A2y + A2z
~r = xî + y ĵ + z k̂
~r2 − ~r1
∆~r
~vav =
=
t2 − t1
∆t
∆~r
d~r
~v = lim
=
∆t→0 ∆t
dt
dy
dz
dx
, vy = , vz =
vx =
dt
dt
dt
~v2 − ~v1
∆~v
~aav =
=
t2 − t1
∆t
d~v
∆~v
~a = lim
=
∆t→0 ∆t
dt
dvx
dvy
dvz
ax =
, ay =
, az =
dt
dt
dt
~ =A
~ +B
~ =B
~ +A
~
R
Rx = Ax + Bx , Ry = Ay + By , Rz = Az + Bz
~ cos φ
~ B
~ ·B
~ = AB cos φ = A
A
~ ·B
~ = Ax Bx + Ay By + Az Bz
A
~ =A
~ × B,
~
C
C = AB sin φ
Cx = Ay Bz − Az By ,
Cz = Ax By − Ay Bx
Cy = Az Bx − Az Bz ,
Simple motions
Constant acceleration in one direction:
v = v0 + at
1
x = x0 + v0 t + at2
2
v 2 = v02 + 2a(x − x0 )
v0 + v
x − x0 =
t
2
Projectile motion:
x = (v0 cos α0 )t
1
y = (v0 sin α0 )t − gt2
2
v = v0 cos α0
vy = v0 sin α0 − gt
Uniform circular motion:
v2
4π 2 R
arad =
= ω2R =
R
T2
Force and Momentum
X
~ = m~a,
F
~ A on B = −F
~ B on A
F
w = mg
fk = µk n, fs 6 µs n
X
~
dP
~ ext
=
F
dt
Zt2 X
~ dt,
~J = ~p2 − ~p1 =
F
t1
M=
X
mi
i
X
~ ext = M~acm
~ = d~p ,
F
F
dt
~ = m1~v1 + m2~v2 + m3~v3 + . . . = M~vcm
P
P
mi~ri
m1~r1 + m2~r2 + m3~r3 + . . .
i
~rcm =
=
M
m1 + m2 + m3 + . . .
~p = m~v,
X
CC1529
Semester 1, 2015
Page 2 of 3
Work and Energy
Rotational Motion
1
1
K = mv 2 , Ugrav = mgy, Uel = kx2
2
2
F = −kx
Wtot = K2 − K1 = ∆K
~ · ~s = F s cos φ
W =F
Z P2
Z P2
~ · d~l
F
F cos φ dl =
W =
dθ
,
v = rωz
dt
d2 θ
dωz
= 2
αz =
dt
dt
2
dv
dω
v
= ω 2 r, atan =
=r
= rαz
arad =
r
dt
dt
IP = Icm + M d2 , vcm = Rω
X
I = m1 r12 + m2 r22 + . . . =
mi ri2
P1
P1
∆W
Pav =
∆t
dW
~ · ~v
P =
=F
dt
Wel = −∆Uel ,
Wgrav = −∆Ugrav
E =K +U
∆E = Wother
Periodic Motion
ω
1
2π
, f=
=
ω = 2πf =
T
2π
T
r
r
k
κ
ω=
, ω=
m
I
r
r
g
mgd
ω=
, ω=
L
I
Fx = −kx
x = A cos(ωt + φ)
1
1
1
E = mvx2 + kx2 = kA2 = constant
2
2
2
r
k
b2
−(b/2m)t
0
0
x = Ae
cos ω t, ω =
−
m 4m2
√
bcritical = 2 k m
Fmax
A= q
2
(k − mωd2 ) + b2 ωd2
ωz =
i
~
~ = ~r × F
τ = rF sin θ, τ
X
X
~
dL
~ =
τz = Iαz ,
τ
dt
1
1
2
+ Icm ωz2 , P = τz ωz
K = M vcm
2
2
Z θ2
1
1
W =
τz dθ, Wtot = Iω22 − Iω12
2
2
θ1
~ = ~r × ~p = ~r × m~v (particle)
L
~ = Iω
~
L
(rigid body)
Moments of inertia
1
M L2
12
1
Thin rod, axis through one end:I = M L2
3
Thin rod, axis through centre:I =
1
M (a2 + b2 )
12
1
Thin rectangular plate, axis along edge:I = M a2
3
1
Hollow cylinder:I = M (R12 + R22 )
2
1
Solid cylinder:I = M R2
2
Thin-walled hollow cylinder:I = M R2
2
Solid sphere:I = M R2
5
2
Thin-walled hollow sphere:I = M R2
3
Rectangular plate, axis through centre:I =
CC1529
Semester 1, 2015
Page 3 of 3
Thermal physics
∆L = αL0 ∆T, ∆V = βV0 ∆T
Q = mc∆T, Q = nC∆T, Q = ±mL
pV = nRT = N kT
N = nNA
3
CV = R (ideal monatomic gas)
2
5
CV = R (ideal diatomic gas)
2
CV = 3R (ideal monatomic solid)
CP
CP = CV + R, γ =
CV
f
CV = R (f = degrees of freedom)
2
γ
pV = constant
T V γ−1 = constant
r
3RT
vrms =
M
mtot = nM = nNA m
QC W
e=
= 1 − QH
QH K=
|QC |
|QC |
=
|W |
|QH | − |QC |
Mechanical waves
v = λf,
s
1
2L
F
µ
β = (10 dB) log
I
I0
∆U = Q − W
dU = dQ − dW (infinitesimal process)
Z 2
dQ
∆S =
(reversible process), S = k ln w
T
1
P
I=
A
dQ
TH − TC
H=
= kA
, Hnet = Aeσ(T 4 − Ts4 )
dt
L
dQabs
Habs =
= AeσTs 4 and Habs = aIA
dt
dQrad
= AeσT 4
Hrad =
dt
λpeak T = 2.898 × 10−3 m.K
∆U = nCV ∆T
Q = nCV ∆T, Q = nCp ∆T
Q = W = nRT ln (V2 /V1 )
TH − TC
TC
=
eCarnot = 1 −
TH
TH
TC
KCarnot =
TH − TC
3
1
3
Ktr = nRT,
m(v 2 )av = kT
2
2
2
v=
F
µ
y(x, t) = A cos(kx ± ωt)
y(x, t) = (ASW sin kx) sin ωt (standing wave)
String fixed at both ends:
v
fn = n
= nf1 (n = 1, 2, 3, . . .)
2L
s
f1 =
p dV,
V1
s
v=
V2
W =
Longitudinal sound waves
2π
k=
λ
ω = 2πf = vk,
Z
r
B
ρ
s
(fluid), v =
Y
ρ
(solid rod)
γRT
(ideal gas)
M
nv
fn =
(n = 1, 2, 3, . . .) (open pipe)
2L
nv
fn =
(n = 1, 3, 5, . . .) (stopped pipe)
4L
v + vL
v
fL =
fs ,
sin α =
v + vS
vS
fbeat = |fa − fb |
v=