Physics Equation Sheet

EQUATIONS IN MOTION
Average speed
v=

d
t

d = distance, t = time

Average velocity
x
v=
t

x = displacement, t = elapsed time

Average acceleration

v = change in velocity, t = elapsed

time

a=

v
t

Linear motion kinematics 1-D

(constant acceleration a)

v = v0 + at
x = x0 + v0t + 12 at 2
v 2 = v02 + 2a( x x0 )
v = 2gh (Free fall from 0 velocity)

To apply in two dimensions, the easiest

way is to choose an x-y coordinate
system so that the direction of the
acceleration is entirely along either the x
or the y direction. This greatly simplifies
things as the acceleration in the other
coordinate direction will have a
component of 0 and the motion in that
other direction will have constant
velocity. The components of motion in
the x and y directions are analyzed
separately.

FG = G

G = 6.6710-11 Nm2/kg2

m1m2
r2

Inclined Planes

F incline = mg sin

is the angle between the inclined plane and

Fnormal = mg cos

the horizontal surface

Hookes Law

The further a spring is stretched, the more force

it pulls back with.

Torque

Torque, which can be roughly thought of as a

twisting force, is proportional to the force
applied and the lever arm length.

F = kx

= Fl

WORK AND ENERGY

Work done by a constant
force

W = Fd cos

Work W done by a constant force of

magnitude F on an object as it is displaced by
a distance d. The angle between the directions
of F and d is

Vector components

v x = v cos ,

v = v x2 + v y2

v y = v sin

tan =

vy

For a vector of magnitude v making an

angle with the x-axis

vx

Centripetal acceleration

v2
aR =
R

Kinetic energy
Centripetal acceleration aR is toward the
center of the circle of radius r for an
object traveling with constant speed v

FORCES AND TORQUE

Newtons first law of
motion (Equilibrium)

At equilibrium, every body continues in its state

of rest or of uniform speed as long as no net
force and no net torque act on it.

Newtons second law of

motion (Dynamics)

The acceleration a of an object is directly

proportional to the net force acting on it and is
inversely proportional to its mass. The direction
of the acceleration is in the direction of the net
force action the object.
Whenever one object exerts a force on a
second object, the second exerts an equal and
opposite force on the first.
Opposes any impending relative motion
between two surfaces, where the magnitude
can assume any value up to a maximum of sFN
where s is the coefficient of static friction and
FN is the magnitude of the normal force.
Force between two surfaces sliding against one
another that opposes the relative motion of the
two surfaces, where k is the coefficient of
kinetic friction.
The force FG between two objects of masses m1
and m2 and separated by a distance r. The
value of the universal gravitation constant is:

r
F = 0
clockwise = counterclockwise

F = ma

Newtons third law of

motion
Force of static friction

F fr s FN

Force of kinetic friction

F fr = k FN

Force of gravity between

any two objects

Prepared By : Fayzan Ahmed

BS / MSc Physics Gold Medalist
M.Phil Astrophysics*

K = 12 mv 2

Work is positive if the object is displaced in the

direction of the force and negative if it is
displaced against the force. The work is zero if
the displacement is perpendicular to the
direction of the force.
Kinetic energy K for a mass m traveling at a
speed v.

Gravitational potential
energy

Potential energy U is the energy that an object

of mass m has by virtue of its position relative
to the surface of the earth. That position is
measured by the height h of the object relative
to an arbitrary zero level.

Conservative forces

Gravitational force

Elastic spring
force

Electric force
Non-conservative forces

Frictional forces

Air resistance

Tension

Normal force

Propulsion of a
motor
Conservation of
Mechanical Energy (Only
holds true if non-conservative
forces are ignored)

A force is conservative if either:

The work done by the force on an

object moving from one point to
another depends only on the initial
and final positions and is
independent of the particular path
taken.

The net work done by the force on

an object moving around any
closed path is zero

U = mgh (local )
GMm
U =
( general )
r

E 2 = E1

The total mechanical energy of a system,

remains constant as the object moves,
provided that the net work done by external
non-conservative forces (such as friction and
air resistance) is zero.

K 2 + U 2 = K1 + U 1
Work-energy Theorem

Wnc = K + U + Ei
Rest Mass Energy

E = mc 2

The work due to non-conservative forces Wnc

is equal to the change in kinetic energy K
plus the change in gravitational potential
energy U plus any changes in internal energy
due to friction.
The energy inherent to a particle by nature of it
having a mass.

Power

W
P=
= Fv
t

Buoyant Force

where is the angle between F and v.

Q = Av

P = Fv cos

Linear momentum:

Linear momentum p is the product of an

objects mass m and velocity v. Linear
momentum is a vector that points in the
same direction of the velocity.

p = mv
r
r
p = mv

Principle of conservation of
linear momentum:

Impulse-momentum theorem:

p = F t

Elastic collisions:

Bodies do not stick

together

Kinetic energy is
conserved

Momentum is
conserved
Inelastic collisions:

Bodies stick together if

completely inelastic

Kinetic energy is not

conserved

Momentum is
conserved
Center of Mass (CM or CofM)

xcm =

A completely inelastic collision is one in

which the bodies stick together completely
after the collision.
In inelastic collisions, kinetic energy is not
conserved as some of the initial kinetic
energy is converted into other forms of
energy such as thermal and sound energy.

Wave Velocity

The velocity of a wave is the product of its

frequency and wavelength.

v = f

Wave Period

T=

1
f

Sound decibels

= 10 log

A difference of 10 in decibels corresponds to

sound intensity levels that differ by a factor of 10.
For example, 90dB is 10 times as loud as 80dB.

I
Io

Standing Waves
Both ends fixed or free

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

L=

nn
(n = 1,3,5,...)
4

Beat frequency

m1 x1 + m2 x2
m1 + m2

Density

m
=
V
Pressure

F
( general definition )
A

When a standing wave is formed on a piece of

string, the string length is some fractional multiple
of the standing wave wavelength. Depending on
how the string is fixed, each end can be a node
or an anti-node.

One end fixed one end

free

FLUIDS AND SOLIDS

P=

Strain

WAVES AND PERIODIC MOTION

L=

In a collision, the velocity of the center of

mass of all the colliding objects remains
constant.

For two bodies:

xcm =

r
F

The center of mass is a point that

represents the average location for the
total mass of the system.

M total

Modulus =

An impulse produces a change in an

objects momentum. Impulse is given by
(F) and
the product of average force
the time interval t (t) over which the force
is applied. Impulse is a vector that points
in the same direction as the average force.

r
r
r
p 2 p1 = Fnet t

mi xi

= Vg

The total linear momentum of an isolated

system remains constant.

r
r
P2 = P1

i =1

The buoyant force on an object in fluid is upward

and equal to the weight of the fluid that the object
buoyant
displaces.
Continuity Equation
The volume flow rate of a fluid is proportional to
the cross-sectional area of the pipe and the
velocity of the fluid. Qin must be equal to Qout.
Bernoullis Equation
One way to remember the Bernoulli equation is to
think of it as an energy conservation equation.
1
p + gy + v 2 = constan The three terms roughly correspond to pressure
2
energy, potential energy, and kinetic energy,
respectively.
Elastic modulus of a
A high modulus material is hard and rigid.
solid
Examples are metal and ceramic. A low modulus
material is elastic, like rubber.
Stress

www.prep101.com

MOMENTUM

P = gy

Power P is defined as the rate at which work is

done. It can also be expressed in terms of the
force F being applied to the object traveling at
a speed v. It is more correct to express this
version of the relationship as

Density of a liquid at rest. Density can also be

measured relative to water, which is termed
specific gravity. A specific gravity > 1 means the
liquid is more dense than water, A specific gravity
< 1 means the liquid is less dense than water
The hydrostatic pressure on a fluid volume is
dependent on its depth, and is equal in all
directions.

Hydrostatic pressure at
a fixed depth

Prepared By : Fayzan Ahmed

BS / MSc Physics Gold Medalist
M.Phil Astrophysics*

f beat = f 1 f 2

Doppler effect

f v
=
fs c

v
c

When two waves of constant amplitude but

different frequencies interfere with each other,
the resulting waves amplitude is confined to an
envelope with some periodicity. The frequency of
the envelope is the beat frequency and can be
heard as distinct beats because of the amplitude
variation with time.
The apparent frequency of the source is
increased as the source approaches the
observer, and is decreased as the sources
leaves the observer.

ELECTROSTATICS AND MAGNETISM

Bolztmanns constant k and has a value of: k = 9.0109 Nm/C2
Coulombs law (electric
The magnitude of the force F between two
force)
charges (Q1 and Q2) in terms of the distance r
between the two charges. The direction of the
Q1Q2
force is directed along the line between the two
F =k 2
r
forces. This force is repulsive if the two charges
are both positive or both negative, and attractive

Electric field due to a

point charge q at a
distance r

E=k

if the one charge is positive and the other

negative.
E is a vector and points away from a positive
charge and toward a negative charge.

Q
r2

Electric potential energy

QQ
U =k 1 2
r
Electric potential

The potential energy stored between the

interaction between two point charges.

The electric potential V due to a point charge q

at a distance r away from the charge.

Q
V =k
r
In constant electric fields

r
r
F = qE U = qEd
V = Ed U = Vq

Note that the force F is in the same direction as

the electric field E if the charge q is positive and
in the opposite direction if the charge is
negative.
The energy gained by some charge in a field is
simply force times the distance traveled.
Potential is the energy per unit charge.

Force on a charge moving

in a magnetic field

r
r r
F = qv B
F = qvB sin

r
with
B
r
r
a velocity v experiences a force F . The
magnitude of this force can also be expressed
r
r
in terms of the angle between v and B .
A charge q moving in a magnetic field

V = IR

Resistance of a wire

R=

L
A

Electric power

V2
P = IV = I 2 R =
R
RMS voltage and current (AC
circuits)

Vrms =

V0

I rms =

I0

2
2

The potential difference V across a device

is given by its resistance R and the current I
that flows through it
The resistance R of a length L of wire with a
cross-sectional area A and resistivity .
Resistivity has units m.

The root-mean-square values can be

calculated from the peak values (V0 and I0)
and are used to calculate the average
power

in AC circuits:

P=I

2
rms

2
Vrms
R=
R

For more than two resistances in series:

Req = R1 + R2

Req = R1 + R2 + R3 + R4 + ...

Resistances in parallel

For more than two resistances in parallel:

Capacitance

Q
C=
V

1
1
1
=
+
C S C1 C 2

1
1
1
1
1
=
+
+
+
+ ...
C S C1 C 2 C 3 C 4

C P = C1 + C 2

C P = C1 + C 2 + C3 + C 4 + ...

Electric energy stored by a

capacitor

Amount of electric energy stored in a

capacitor is given in terms of the
capacitance C and the potential difference
between the conductors V.

www.prep101.com
Q

U E = 12 CV 2 = 12 QV = 12

1
1
1
1
1
= +
+
+
+ ...
Req R1 R2 R3 R4
A higher capacitance capacitor can store
more charge at the same voltage.

Prepared By : Fayzan Ahmed

BS / MSc Physics Gold Medalist
M.Phil Astrophysics*

LIGHT AND GEOMETRICAL OPTICS

Snells law

n1 sin 1 = n2 sin 2

Total internal reflection

sin c =

n2
n1

Energy of one photon

E = hf

n=

c
v

The lens equation

1
1 1
+ =
do di
f

With help from Ohms law, electric power P

can be calculated using any combination of
two of the following quantities: resistance R,
voltage V or current I

Resistances in series

1
1
1
= +
Req R1 R2

For more than two capacitors:

Index of refraction

ELECTRONIC CIRCUITS
Ohms law

Capacitors in series CS and

parallel CP

Lateral magnification

m=

hi
d
= i
ho
do

The angle of incidence 1 is with respect to

the perpendicular of the surface between the
two media (with indices of refraction n1 and
n2). The angle of refraction 2 is also with
respect to the perpendicular.
The critical angle c is the angle of incidence
beyond which total internal reflection occurs.
The index of refraction for the medium in
which the incident ray is traveling is n1
The energy of light is dependent on its
frequency. H is the planck constant
6.626068 10-34 m2 kg / s
The higher the index of refraction is for a
medium, the slower is the speed of light in
that medium.
The focal length of the lens f is:

Positive for a converging lens

Negative for a diverging lens

The object distance do is:

Positive if it is on the side of the

lens from which the light is coming

Negative if on the opposite side

The image distance di is:

Positive if it is on the opposite side

of the lens from which the light is
coming

Negative if on the same side

For an upright image, the magnification m is
positive and for an inverted image m is
negative.

Power of a lens

P=

1
f

Focal length of a spherical

mirror

f =

1
r
2

For a spherical mirror, the focal length is half

of the radius of curvature.

APP19 table of elements.qxd

8/13/05

1:52 PM

Page APP 115

APP 115

TABLE OF ELEMENTS

Atomic
Number

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54

Element

Hydrogen
Helium
Lithium
Beryllium
Boron
Carbon
Nitrogen
Oxygen
Fluorine
Neon
Sodium
Magnesium
Aluminum
Silicon
Phosphorus
Sulfur
Chlorine
Argon
Potassium
Calcium
Scandlum
Titanium
Vanadium
Chromium
Manganese
Iron
Cobalt
Nickel
Copper
Zinc
Gallium
Germanium
Arsenic
Selenium
Bromine
Krypton
Rubidium
Strontium
Yttrium
Zirconium
Niobium
Molybdenum
Technetium
Ruthenium
Rhodium
Palladium
Silver
Cadmium
Indium
Tin
Antimony
Tellurium
Iodine
Xenon

Symbol

Atomic
Weight

Atomic
Number

H
He
Li
Be
B
C
N
O
F
Ne
Na
Mg
Al
Si
P
S
Cl
Ar
K
Ca
Sc
Ti
V
Cr
Mn
Fe
Co
Ni
Cu
Zn
Ga
Ge
As
Se
Br
Kr
Rb
Sr
Y
Zr
Nb
Mo
Te
Ru
Rh
Pd
Ag
Cd
In
Sn
Sb
Te
I
Xe

1.00794
4.002602
6.941
9.012182
10.811
12.011
14.00674
15.9994
18.9984032
20.1797
22.989768
24.3050
26.981539
28.0855
30.973762
32.066
35.4527
39.948
39.0983
40.078
44.955910
47.867
50.9415
51.9961
54.93805
55.845
58.93320
58.6934
63.546
65.39
69.723
72.61
74.92159
78.96
79.904
83.80
85.4678
87.62
88.90585
91.224
92.90638
95.94
97.9072*
101.07
102.90550
106.42
107.8682
112.411
114.818
118.710
121.760
127.60
126.90447
131.29

55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107

Element

Cesium
Barium
Lanthanum
Cerium
Praseodymium
Neodymium
Promethium
Samarium
Europium
Gadolinium
Terbium
Dysprosium
Holmium
Erbium
Thulium
Ytterbium
Lutetium
Hafnium
Tantalum
Tungsten
Rhenium
Osmium
Iridium
Platinum
Gold
Mercury
Thallium
Lead
Bismuth
Polonium
Astatine
Radon
Francium
Radium
Actinium
Thorium
Protactinium
Uranium
Neptunium
Plutonium
Americium
Curium
Berkelium
Californium
Einsteinium
Fermium
Mendelevium
Nobelium
Lawrencium
Unnilquadium
Unnilpentium
Unnilhexium
Unnilseptium

Symbol

Atomic
Weight

Cs
Ba
La
Ce
Pr
Nd
Pm
Sm
Eu
Gd
Tb
Dy
Ho
Er
Tm
Yb
Lu
Hf
Ta
W
Re
Os
Ir
Pt
Au
Hg
Tl
Pb
Bi
Po
At
Rn
Fr
Ra
Ac
Th
Pa
U
Np
Pu
Am
Cm
Bk
Cf
Es
Fm
Md
No
Lr
Unq
Unp
Unh
Uns

132.90543
137.327
138.9055
140.115
140.90765
144.24
144.9127*
150.36
151.965
157.25
158.92534
162.50
164.93032
167.26
168.93421
173.04
174.967
178.49
180.9479
183.84
186.207
190.23
192.217
195.08
196.96654
200.59
204.3833
207.2
208.98037
208.9824*
209.9871*
222.0176*
223.0197*
226.0254*
227.0278*
232.0381*
231.0388*
238.0289
237.0482*
244.0642*
243.0614*
247.0703*
247.0703*
251.0796*
252.083*
257.0951*
258.10*
259.1009*
262.11*
261.11*
262.114*
263.118*
262.12*