CC1531 Semester 1, 2015 Page 1 of 3

PHYS1901 Physics 1 (Advanced) Formula Sheet

Vectors~A = Axi + Ay j + Azk

A =∣∣∣~A∣∣∣ =

√A2

x + A2y + A2

z

~R = ~A + ~B = ~B + ~A

Rx = Ax + Bx, Ry = Ay + By, Rz = Az + Bz

~A · ~B = AB cos φ =∣∣∣~A∣∣∣ ∣∣∣~B∣∣∣ cos φ

~A · ~B = AxBx + AyBy + AzBz

~C = ~A× ~B, C = AB sin φ

Cx = AyBz − AzBy, Cy = AzBx − AzBz,

Cz = AxBy − AyBx

Kinematics~r = xi + yj + zk

~vav =~r2 −~r1

t2 − t1=

∆~r

∆t

~v = lim∆t→0

∆~r

∆t=

d~r

dt

vx =dx

dt, vy =

dy

dt, vz =

dz

dt

~aav =~v2 − ~v1

t2 − t1=

∆~v

∆t

~a = lim∆t→0

∆~v

∆t=

d~v

dt

ax =dvx

dt, ay =

dvy

dt, az =

dvz

dt

Mechanics∑~F = m~a, w = mg, ~FA on B = −~FB on A

fk = µkn, fs 6 µsn

~J = ~p2 − ~p1 =

t2∫t1

∑~F dt, M =

∑i

mi

~p = m~v,∑

~F =d~p

dt,

∑~Fext = M~acm

~P = m1~v1 + m2~v2 + m3~v3 + . . . = M~vcm

~rcm =

∑i

mi~ri∑i

mi

=m1~r1 + m2~r2 + m3~r3 + . . .

m1 + m2 + m3 + . . .

Simple motionsConstant acceleration in one direction:v = v0 + at

x = x0 + v0t +1

2at2

v2 = v20 + 2a(x− x0)

x− x0 =

(v0 + v

2

)t

Projectile motion:x = (v0 cos α0)t

y = (v0 sin α0)t−1

2gt2

v = v0 cos α0

vy = v0 sin α0 − gt

Uniform circular motion:

arad =v2

R= ω2R =

4π2R

T 2

CC1531 Semester 1, 2015 Page 2 of 3

Work and EnergyW = ~F ·~s = Fs cos φ

K =1

2mv2, U = mgy

Wtot = K2 −K1 = ∆K

W =

∫ x2

x1

Fx dx

W =

∫ P2

P1

F cos φ dl =

∫ P2

P1

F‖ dl =

∫ P2

P1

~F · d~l

Pav =∆W

∆t

P = lim∆t→0

∆W

∆t=

dW

dt= ~F · ~v

Wel =1

2kx2

1 −1

2kx2

2 = −∆Uel

Wgrav = mgy1 −mgy2 = −∆Ugrav

E = K + U

Fx(x) = −dUx

dx

~F = −(

∂U

∂xi +

∂U

∂yj +

∂U

∂zk

)

Periodic Motion

ω = 2πf =2π

T, f =

ω

2π=

1

T

ω =

√k

m, ω =

√κ

I

ω =

√g

L, ω =

√mgd

I

Fx = −kx, ax =Fx

mx = A cos(ωt + φ)

E =1

2mv2

x +1

2kx2 =

1

2kA2 = constant

x = Ae−(b/2m)t cos ω′t, ω′ =

√k

m− b2

4m2

bcritical = 2√

k m

A =Fmax√

(k −mω2d)

2+ b2ω2

d

Rotational Motion

ωz = limx→0

∆θ

∆t=

dθ

dt

αz = limx→0

∆ωz

∆t=

dωz

dt=

d2θ

dt2

arad =v2

r= ω2r, atan =

dv

dt= r

dω

dt= rαz

v = rωz, IP = Icm + Md2, vcm = Rω

I = m1r21 + m2r

22 + . . . =

∑i

mir2i

τ = F l = rF sin θ, ~τ = ~r× ~F∑τz = Iαz,

∑~τ =

d~L

dt

K =1

2Mv2

cm +1

2Icmω2

z , P = τzωz

W =

∫ θ2

θ1

τz dθ, Wtot =1

2Iω2

2 −1

2Iω2

1

~L = ~r× ~p = ~r×m~v (particle)~L = I~ω (rigid body)I = MR2 (thin hollow cylinder)

I =1

2MR2 (solid cylinder)

I =2

5MR2 (solid sphere)

Newtonian Gravitation

Fg =Gm1m2

r2, g =

GmE

R2E

w = Fg =GmEm

R2E

, v =

√GmE

r

U = −GmEm

r, RS =

2GM

c2

T =2πr

v=

2πr3/2

√GmE

CC1531 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 = NkT N = nNA

CV =3

2R (ideal monatomic gas)

CV =5

2R (ideal diatomic gas)

CV = 3R (ideal monatomic solid)

CP = CV + R, γ =CP

CV

vrms =

√3RT

M

mtot = nM = nNAm

e =W

QH

= 1 +QC

QH

= 1−∣∣∣∣QC

QH

∣∣∣∣eCarnot = 1− TC

TH

=TH − TC

TH

KCarnot =TC

TH − TC

eOtto = 1− 1

rγ−1

Ktr =3

2nRT,

1

2m(v2)av =

3

2kT

Mechanical waves

v = λf, k =2π

λ

ω = 2πf = vk, 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:

fn = nv

2L= nf1 (n = 1, 2, 3, . . .)

f1 =1

2L

√F

µ

β = (10 dB) logI

I0

W =

∫ V2

V1

p dV, ∆U = Q−W

dU = dQ− dW (infinitesimal process)

K =|QC ||W |

=|QC |

|QH | − |QC |

∆S =

∫ 2

1

dQ

T(reversible process), S = k ln w

H =dQ

dt= kA

TH − TC

L, Hnet = Aeσ(T 4 − T 4

s )

W = nCV (T1 − T2)

=CV

R(p1V1 − p2V2) (adiabatic process, ideal gas)

=1

γ − 1(p1V1 − p2V2)

W = nRT ln

(V2

V1

)

Reversible Processes for Ideal Gases:Adiabatic (no heat transfer):

Q = 0, pV γ = constantIsochoric (constant volume): W = 0

Isobaric (constant pressure): W = p(V2 − V1)

Isothermal (constant temperature)

Longitudinal sound waves

v =

√B

ρ(fluid), v =

√Y

ρ(solid rod)

v =

√γRT

M(ideal gas)

fn =nv

2L(n = 1, 2, 3, . . .) (open pipe)

fn =nv

4L(n = 1, 3, 5, . . .) (stopped pipe)

fL =v + vL

v + vS

fs, sin α =v

vS

fbeat = |fa − fb|