Force due to spring is given by:
Use FBD for spring problems, the sign of the force should be clear from the diagram
Equation sheet Exam 1 PHY2048
Newton’s Laws of Motion
1. A body at rest will remain at rest a body in motion will remain in motion
unless acted upon by an external force
2. The net sum of forces accelerates an object by an amount proportional to
its mass and in the direction of the net forces.
3. For every action there is an opposite and equal reaction. Action reaction
pairs never act on the same object
mF = a
0F =
A on B B on A F F
Force of friction comes in two flavors. Static frictional forces apply when the object is at rest
with respect to the surface. Kinetic frictional forces apply when the object is moving with
respect to the surface. Both frictional forces always act parallel to the surface and are
proportional to the normal force.
2 2
1tan
x x x
y y y
x y
y
x
=
A B C
A B C
C C
C
C
A+ B C
C
2 2
1tan
x x x
y y y
x y
y
x
=
A B C
A B C
C C
C
C
A B C
C
B
Chapter 1: Measurements, Estimation, Vectors
Chapter 4&5: Newton’s Law and Applications
Definition of velocity and acceleration. note: velocity and acceleration are vector
quantities
0
2
0 0
2 2
0 0
1
2
2
fx x x
f x x
fx x x f
v v a t
x x v t a t
v v a x x
2
rad
va
R
For motion in a circle the acceleration vector points towards
the center of the circle with a magnitude given by:
Equations of motion in 2 or 1 dimension are given below. These apply only when the
acceleration is constant, gravitational acceleration is an example of a constant a. The
time t is the quantity that is common to both dimensions for 2-D problems
Chapter 2&3: Motion in a Straight Line and in a Plane
0
0
, lim
, lim
f i f ii
average instt
f i f i
average instt
t t dt
t t dt
x x x x dxv v
v v v v dva a
friction k
friction s
F N
F N
AC
B
Distance: A “scalar” quantity that describes the length of the path taken between
two locations
Displacement: A “vector” that points between two locations. The vector begin at theinitial location and ends at the final.
i
i
dist s
f i x = x xA
B
C
|A|
|A|cos
|A|sin
0
2
0 0
2 2
0 0
1
2
2
fy y y
f y y
fy y y f
v v a t
y y v t a t
v v a y y
( |0) |0
( |0) |0
0
cos
sin
y
x
f x f
f y f
a g
a
v v
v v
For projectile motion in the plane
v
x
y
Spring k F x
0 0 0
0 0 0
0
0
( ) ( )
( ) ( )
v t t
v t t
x t t
x t t
d t dt t dt
d t dt t dt
v a v = v a
x v x = x v
𝑣
equations of motion that apply generally
Equation sheet Exam II
Chapter 8: Momentum, Impulse, Conservation of Momentum
2
1
, ,
: ,
Momentum is conserved if there are no external forces
0
Impulse is defined as:
Impulse-momentum theorem:
x x y y
ext T initia
t
t
l T final
m p mv p mv
t
dt
p v
pF = p p
J = F
J = 12 p p
Collisions (momentum is cnrv’d)
Elastic: KE is conserved implies
the following velocity relation
Inelastic: KE is not conserved
Completely inelastic: objects stick
together after they collide and
1 1 2 2 3 3
1 2 3
1 1 2 2 3 3
1 2 3
cm
cm
x m x m x mx
m m m
y m y m y my
m m m
Center of Mass and Momentum
, , , ,B f A f B i A i v v v v
, ,B f A fv v
2 2
1 1
2 2
2 1 2 1
Work definition
cos
Work by varing force or curved path
Work-energy relation, definition of
1 1
2
cos
2
s s
s s
W Fs
W
KE
W K KE KE m
d
v m
d s
v
F
F s
F s
||
||
Definition of Power
,
,
av av av
WP P Fv
t
dWP P Fv
dt
Chapter 6: Work, Energy and Power
Chapter 10: Rotational Dynamics
The torque is given by the magnitude of where is a vector
pointing from the pivot point to the where the acts. The angle is the smallest angle between
, s
the vect
in ,
r F r F r
F
ors when located tail to tail.
, where is the moment of inertia and is the angular velocity
, for a single particle where is perpendicular distance from
axis a
Angular Mom
nd is
entu
the linear mo
m
mentu
L I
L mv
I
l
mv
l
2 2
2 1
,1 ,2
Work Energy theorem for angular motion
1
m
Conservation of Angular Momenta follows from Newton's 2
(2
thus if and onf if 0, ., the are
nd
no ext
Law
e
)
Total Total
ext
CM
ext
d dI
dt dt
W I
LI
L L ie
rnal torques
, for rigid bodies the relation is useful a rI
Chapter 9: Rotational Motion
2 1
2
0
2 2
1 12 2
2 1
Angular velocity and acceleration
;
Angular kinematic relati
1
on
2
( )
s
2
inst inst
d d
dt dt
t
t t
2 2 2 2
1 1 2 2 3 3
Potential Energy
Moment of Inertia, general form, see table 9.2
...i
CM
i ir r r m
U Mgy
I m rm m
2
2
Angular to linear relations
arc length
tangential
tangential
tange
radial
ntial
r
r
s r
v
a
va r
r
Rolling without slip
,
ping
0, 2bottom CM topv v vR R
2 2
,1 1 2
,2 ,1
,1
,1 ,2 1 2
,2
2
Potential Energy ( ) (conservative forces)
( )
1( )
2
Conservation of Energy
ela
gravity grav grav grav
elasstic elas
Total Total
Total elas grav other
elas
U
W U U mg y
W U U k x x
E E
E KE U U W
U
U
KE
y
,2 ,2 1 ,1 ,1grav elas grav elas otherU U KE U U W
Chapter 7: Potential Energy and Conservation of Energy
Force from Energy is
( ) ( )
In 3-D
ˆ ˆ ˆ
x
dU xF x
dt
U U U
x y x
F i + j + k
2 21 12 2
Kinetic Energy of object with rotational and linear motion
total cm cmKE Mv I
Equillibrium of rigid body
0 : 0, 0 and 0x yF F F
FBD for man w/ladder on friction less wall, The torque is computed
about axis at B but you can use anywhere else as axis of rotation
To solve equilibrium problems
begin by using FBD approach.
Each force is now applied on
extended object. Next write down
S F = 0, in x and y, and then write
down S = 0, take the axis of
rotation for the torques to be
anywhere on the object that is the
most convenient. Finally solve for
the unkwnown(s) from the
resulting three equation.
Chapter 11: Equilibrium
The speed of a mechanical wave is:
, where is frequency and is the wavelength.
The magnitude of depends only on the physical properties
of the media through which the wave is propagating.
wavev f f
v
Thus for a string of
length the wave speed
, where is the linear mass density or mass per unit length of the string.wave
l
Fv
Equation sheet Final Exam, PHY2048
Chapter 15: Mechanical Waves
Equation that describes a mecha
,
,
nical wave
( , ) cos 2
( , ) cos
y x t A
y x t A
x t
T
kx t
These eq. describe waves propagating to the right
with a phase (), amplitude A, angular frequency
𝜔 and wave number k
2
2
2
Velocity and acceleration of any particle on a transverse
( , ) sin
( , )
wa
o
e
s
v
c
y
y
dxv x t A kx
dt
d xa x t A kx
dt
t
t
2 2
2 2 2 2
max a e
2
v
Power in a sinousoidal wave,
( , ) sin
1,
2
P x t A kx t
P A
F
F P F A
1 2
2N
m m
rF = G r
2
22
2
G rad G Sat
Sat Earth EarthN Sat N
vF = ma => F = m
r
m m mvG m G v
r r r
Newton’s Law of Universal Gravity. F is the force exerted by objects of mass
m1 and m2 on one another at a distance r apart. The direction of force is along
the line joining the two objects. These force obey Newton’s third law and can
be considered an action reaction pair.
Equations for satellite motion in a circular orbit. Here r is the distance between the
satellite and the object being orbited. GN is Newton’s constant and T is the period
or time it takes the satellite to go one time around.
Chapter 13: Gravitation
2 2
Potential Energy ( ) of a mass a distance from center of the Earth
The weigth and gravitational constant g at surface of the earth is
, i
N EE
N E N Eg
E E
U r
G m mU
r
G m m G mw F g
R R
Kepler’s Laws
1. Each planet moves in an elliptical orbit with the sun at one of the foci of the ellipse
2. A line from a planet to sun sweeps out equal area in equal times
3. The periods of the planets are proportional to the 3/2 power of the major axis
length of their orbit
2 2
2 2
22
2
Simple harmonic motion, restorative force proportional to
, ,
Equations of motion for a mass attached to a spring
( ) cos( ); sin ; cos ;
x x
xx
x
d x d x kF kx ma kx m kx x
dt dt m
dx d xx t A t v A t a A t
dt dt
k
m
Amplitude ( max displacement from equillibrium)
Period, time it takes to complete one cycle
Frequency, how many cycles per sec Hz or 1
Angular frequency which is equal to 2
1, freque
A m
T s
f s
f rad s
fT
ncy and period are inverse of each other
Chapter 14: Periodic Motion
The physical pendulum
, where is gravitational constant and is the distance btw center-of-mass and axis, is the mass and is the moment of inertiamgd
d m II
g
Angular simple harmonic motion is
, where is the torsion constant and is the moment of inertiaII
2 2 2 2
Total Energy of a mass attached to a spring on a frictionless surface
1 1,
2 2T x
kE mv kx v A x
m
2
3/2
11 2 2
2 2
6.67 10
EarthN
N Earth
N
m rG T r
r T G m
G N m kg
The simple pendulum
, where g is the gravitational acceleration, L is the length of the pendulumg
L
2
1 2
2 2
2 1
Power and Intensity and the Inverse square law
; 4
r
r
PI
I r
I
2 2with and k
T
1 2
Wave superposition
( , ) ( , ) ( , )totaly x t y x t y x t
1
1
( , ) sin sin
( 1,2,3...)2
Standing waves on a string
2 1( 1,2,3...)
SW
n
n
y x t A kx t
vf n nf n
L
Ln
n n
Chapter 12: Fluid Mechanics
Density and press
;
ure
p
m
V
dF
dA
2 2
Pressure in a fluid at rest (Pascal's Law)
, p p gh Where the p2 and p1 are the
pressures at pts 2 and 1, 𝜌 is the density of the fluid, g is
the gravitational acceleration and h is the distance in the
vertical btw pts 2&1
Archimede's principle
F B f gV The Buoyant is theupward forceexcerted by fluid on abody immersedequal to amount offluid displaced.
0
F 0 B mg
F
2 21 11 1 1 2 2 22 2
Bernoulli's equation
= p gy v p gy v
gauge ab atmp p p
Absolute pressure: The total pressure including
atmospheric. Gauge pressure: The excess pressure
above atmospheric pressure