Semester One Examinations, 2022

PHYS1001

PHYS1001: Useful formulae and numerical values

Mathematics

Quadratic eqn:

A< # + C< + D = 0
Solution:
< = Ã¢Ë†â€™C Ã‚Â±
Ã¢Ë†Å¡% ! &'()
#(
Trigonometry
DEF # - + FG># – = 1

sin(J Ã‚Â± K) = sin J cos K Ã‚Â± cos J sin K

cos(J Ã‚Â± K) = cos J cos K Ã¢Ë†â€œ sin J sin K

JÃ‚Â±K

JÃ¢Ë†â€œK

sin J Ã‚Â± sin K = 2 sin M

P

N cos O

2

2

J+K

JÃ¢Ë†â€™K

cos J + cos K = 2 cos M

N cos M

N

2

2

J+K

JÃ¢Ë†â€™K

cos J Ã¢Ë†â€™ cos K = 2 sin M

N sin M

N

2

2

Binomial expansion

(1 + < +
>(> Ã¢Ë†â€™ 1) #

< +Ã¢â€¹Â¯
2!
Calculus
d n
x = n x n -1
dx
x n +1
n
x
dx
=
ÃƒÂ²
n +1
1
ÃƒÂ² x dx = ln ( x )
d
sin ( ax ) = a cos ( ax )
dx
d
cos ( ax ) = - a sin ( ax )
dx
Vectors
UÃ¢Æ’â€”= JK cos V
JÃ¢Æ’â€”Ã¢Ë†â„¢K
UÃ¢Æ’â€” = JK sin V >UÃ¢Æ’â€”

JÃ¢Æ’â€”Ãƒâ€”K

UÃ¢Æ’â€”.

where >UÃ¢Æ’â€” is a unit vector normal to both JÃ¢Æ’â€” and K

Page 9 of 14

Semester One Examinations, 2022

PHYS1001

Vectors Ã¢â‚¬â€œ continued

If XÃ¢Æ’â€”, YÃ¢Æ’â€”, and ZUÃ¢Æ’â€” are mutually perpendicular unit vectors then

UÃ¢Æ’â€” =Z

UÃ¢Æ’â€” Ã¢Ë†â„¢XÃ¢Æ’â€”= 0 and XÃ¢Æ’â€”Ãƒâ€”XÃ¢Æ’â€”=YÃ¢Æ’â€”Ãƒâ€”YÃ¢Æ’â€”=Z

UÃ¢Æ’â€” Ãƒâ€”ZUÃ¢Æ’â€”= 0 and XÃ¢Æ’â€”Ãƒâ€”YÃ¢Æ’â€”=Z

UÃ¢Æ’â€” , YÃ¢Æ’â€”Ãƒâ€”Z

UÃ¢Æ’â€” =XÃ¢Æ’â€”, Z

UÃ¢Æ’â€” Ãƒâ€”XÃ¢Æ’â€”=YÃ¢Æ’â€”.

XÃ¢Æ’â€”Ã¢Ë†â„¢XÃ¢Æ’â€”=YÃ¢Æ’â€”Ã¢Ë†â„¢YÃ¢Æ’â€”=ZUÃ¢Æ’â€”Ã¢Ë†â„¢ZUÃ¢Æ’â€”= 1 and XÃ¢Æ’â€”Ã¢Ë†â„¢YÃ¢Æ’â€”=YÃ¢Æ’â€”Ã¢Ë†â„¢Z

XÃ¢Æ’â€”

Ã¢Æ’â€”

UÃ¢Æ’â€”

JÃƒâ€”K= [J+

K+

Kinematics and Dynamics

vs = ds / dt

as = dvs / dt

w = dq / dt

a = dw / dt

tf

tf

s f = si + ÃƒÂ² vs dt

ti

tf

YÃ¢Æ’â€”

J,

K,

s

r

vt = Ãâ€° r

ÃŽÂ¸=

q f = qi + ÃƒÂ² w dt

ti

tf

v f = vi + ÃƒÂ² as dt

w f = wi + ÃƒÂ² a s dt

v fs = vis + as Dt

w f = wi + a Dt

1

s f = si + vis Dt + as (Dt ) 2

2

2

2

v fs = vis + 2as Ds

1

q f = qi + wi Dt + a (Dt ) 2

2

2

2

w f = wi + 2a Dq

ti

ZUÃ¢Æ’â€”

J- [

K-

at = ÃŽÂ± r

ti

vt2

ar =

r

2Ãâ‚¬ r 2Ãâ‚¬

T=

=

v

Ãâ€°

Impulse, Momentum, Energy, Work

Ã¯ÂÂ²

Ã¯ÂÂ²

Pf = Pi

Ã¯ÂÂ²

Ã¯ÂÂ²

F = d p / dt

tf

ÃŽâ€Esys = ÃŽâ€K + ÃŽâ€U + ÃŽâ€Eth = Wext

K f +U f + ÃŽâ€Eth = K i +U i +Wext

J x = Ã¢Ë†Â« Fx (t)dt

ÃŽâ€K = Wnet = Wc +Wdiss +Wext

ÃŽâ€px = J x

W = Ã¢Ë†Â« Fs ds

K f +U f = K i +U i

ÃŽâ€Eth = f k ÃŽâ€s

Ã¯ÂÂ² Ã¯ÂÂ²

W = F Ã¢â‚¬Â¢ ÃŽâ€r

ti

1

K = mv 2

2

U g = mgy

1

U s = k(ÃŽâ€s) 2

2

Fs = Ã¢Ë†â€™kÃŽâ€s

sf

si

Fs = Ã¢Ë†â€™dU / ds

P=

dEsys

Ã¯ÂÂ²dt Ã¯ÂÂ²

P = F Ã¢â‚¬Â¢v

Page 10 of 14

Ã¯ÂÂ²

Ã¯ÂÂ² Fnet

a=

m

FG = mg

f s,max = Ã‚Âµ s n

f k = Ã‚Âµk n

f r = Ã‚Âµr n

DÃ¢â€°Ë†

1 2

Av

4

Semester One Examinations, 2022

PHYS1001

Rigid Body Rotation

ÃŽÂ±=

Ã¯ÂÂ²

Ã¯ÂÂ²

L = IÃâ€°

Ã¯ÂÂ²

dL

= Ãâ€ž net

dt

1

xcm =

Ã¢Ë†Â« x dm

M

1

ycm =

Ã¢Ë†Â« y dm

M

I = Ã¢Ë†â€˜ mi ri 2 = Ã¢Ë†Â« r 2 dm

Ãâ€ž net

I

1

1

2

E = K rot + K cm +U g = IÃâ€° 2 + Mvcm

+ Mgycm

2

2

! ! !

Ãâ€ž =rÃƒâ€”F

Ãâ€ž = rF sin ÃŽÂ¸ = rFt = dF

vcm = RÃâ€°

I = I cm + Md 2

i

Moments of Inertia

1

ML2

12

1

I = ML2

3

1

I = Ma 2

12

1

I = Ma 2

3

I=

Thin rod, about centre.

Thin rod, about end.

Plane or slab, about centre.

Plane or slab, about edge.

1

MR 2

2

I = MR 2

2

I = MR 2

5

2

I = MR 2

3

I=

Cylinder or disk, about centre.

Cylindrical hoop, about centre.

Solid sphere, about diameter.

Spherical shell, about diameter.

Oscillations

( Fnet ) s = – ks

w=

k

m

T = 2p

m

k

x(t ) = A cos(wt + f0 )

ÃƒÂ¦ mg ÃƒÂ¶

( Fnet )t = – ÃƒÂ§

ÃƒÂ·s

ÃƒÂ¨ L ÃƒÂ¸

w=

g

L

T

= 2p

vx (t ) = -vmax sin(wt + f0 )

L

g

1 2 1 2 1

1

mvx + kx = m(vmax ) 2 = kA2

2

2

2

2

– t /t

E = E0 e

E=

f = 1/ T

w = 2p f = 2p / T

ax = -w 2 x

vmax = w A

x(t ) = Ae – bt / 2 m cos(wt + f0 )

t = m/b

Page 11 of 14

Semester One Examinations, 2022

PHYS1001

Fluids and Elasticity

r = m /V

p=F/A

p = p0 + r gh

v1 A1 = v2 A2

1

1

p1 + r v12 + r gy1 = p2 + r v22 + r gy2

2

2

Thermodynamics

ÃŽâ€Eth = W + Q

pV = nRT

Vf

pV = Nk BT

W =Ã¢Ë†â€™Ã¢Ë†Â«

p2V2 p1V1

=

T2

T1

ÃŽÂ³ = C P / CV

M

m

N M (in grams)

n=

=

NA

M mol

Q = Ã‚Â±ML

Q = Mc ÃŽâ€T

Q = nC ÃŽâ€T

Q / ÃŽâ€t = (kA / L) ÃŽâ€T

Vi

p dV

C P = CV + R

N=

Number density = N /V

Q / ÃŽâ€t = eÃÆ’ AT 4

9

TF = TC + 32o

5

TK = TC + 273

pV ÃŽÂ³ = const

TV ÃŽÂ³ Ã¢Ë†â€™1 = const

1N

2N

2

p=

mvrms

=

ÃŽÂµ

3V

3 V avg

3

ÃŽÂµavg = k BT

2

3

3

Eth = Nk BT = nRT (Monatomic gas)

2

2

5

5

Eth = Nk BT = nRT

(Diatomic gas)

2

2

Eth = 3Nk BT = 3nRT (Elemental solid)

vrms = (v 2 )avg

TpÃŽÂ³ /(ÃŽÂ³ Ã¢Ë†â€™1) = const

W = Ã¢Ë†â€™nRT ln(V f /Vi )

ÃŽÂ·=

Wout

QH

T

ÃŽÂ· Ã¢â€°Â¤ 1Ã¢Ë†â€™ C

TH

K=

QC

Win

KÃ¢â€°Â¤

TC

TH Ã¢Ë†â€™ TC

Ws = Ã¢Ë†Â« p dV

Page 12 of 14

F

DL

=Y

A

L

DV

p = -B

V

Semester One Examinations, 2022

PHYS1001

Numerical Values

= 9.80 m s&#

^ = 6.672 Ãƒâ€” 10&.. m/ kg &. s&# or N m# kg &#

Earth Mass = 6.0 Ãƒâ€” 10#’ kg

Earth Radius = 6.4 Ãƒâ€” 100 m

Solar Mass = 2.0 Ãƒâ€” 10/1 kg

Solar Radius = 7.0 Ãƒâ€” 102 m

Earth Ã¢Ë†â€™ Sun mean distance = 1.5 Ãƒâ€” 10.. m

one (metric) tonne = 1.0 Ãƒâ€” 10/ kg

Z! = 1.3801 Ãƒâ€” 10&#/ J K &.

p = 5.67 Ãƒâ€” 10&2 W m&# K &’

c = 4190 J kg &. K &. (for water)

r3 = 3.33 Ãƒâ€” 104 J kg &. (for water)

r5 = 22.6 Ãƒâ€” 104 J kg &. (for water)

r5 = 2.00 Ãƒâ€” 104 J kg &. (for carbon dioxide, dry-ice sublimation)

s = 1.6606 Ãƒâ€” 10 kg

t7 = 6.022 Ãƒâ€” 10#/ particles mol&.

v89: (H) = 0.001 kg mol&.

x = 8.314 J mol&. K &.

?” =

/

#

x (monatomic gas)

4

?” = # x (diatomic gas)

y = 1.67 (monatomic gas)

y = 1.40 (diatomic gas)

1 atm = 101.3 kPa = 1.013 Ãƒâ€” 104 Pa

Uncertainty Analysis

Summary of rules for combining uncertainties for dependent measurements

If / = < + { + | + Ã¢â€¹Â¯ or / = < Ã¢Ë†â€™ { Ã¢Ë†â€™ | Ã¢Ë†â€™ Ã¢â€¹Â¯ then
ÃŽâ€/ = ÃŽâ€< + ÃŽâ€{ + ÃŽâ€z + Ã¢â€¹Â¯.
If / = < Ãƒâ€” { Ãƒâ€” | Ãƒâ€” Ã¢â‚¬Â¦ or / =

Purchase answer to see full

attachment