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Chapter 2 - Motion
Part I - KINEMATICS
physics
how objects move
why objects move
how objects interact
(environment)
KINEMATICS - how objects move (not why)
Description of motion - change in position
cars
sports: baseball, football, soccer, etc.
world: rotates and revolves
How to measure position?
PHYSICAL QUANTITIES
describe the physical universe
two types of physical quantities:
SCALARS - described by a magnitude or quantity
how much, how far
just describe amount
mass, time, volume, length, temperature, density, speed
Vectors: magnitude and direction!
quantities for which direction is important
i.e., where? (displacement - distance and direction)
velocity, acceleration, force, momentum, magnetic &electric field
displacement - like directions on map
Difference between scalars and vectors
1 step = 2 ft
20 feet
Distance=
displacement =
How to describe vectors - distance and direction
~
Vector variables - A A A A
A
~
Geometric description - represent with a directed line
scale factor - magnitude  length - ruler
direction - points along arrow - protractor
Scale 1 inch = 1mile
head
A
magnitude =
tail
direction =
VECTOR ALGEBRA - adding vectors
A
+
B
=
C?
head-to-tail method: head of first
to tail of second
result:
C from beginning to end
magnitude :
direction :
displacement from adding displacements
Resolution of Vectors - component description
Break vectors into components
up/down - left/right
scale built in
components give direction
y
A
Ay
m
in
miles
Ax
x
Looks like two
perpendicular rulers
A = A + A
x
y
x-component and y-component specifies vector
easy component directions (perpendicular)
like treasure map
Vector - TWO SCALAR COMPONENTS
Can treat each direction
separately as a vector
Rectilinear Kinematics
MOTION - changes in position
how objects move without regard to why
one-dimensional motion
Kinematic physical quantities-how you move
position, how fast, speed up
SCIENTIFIC MODEL
straight line - start and end
point particle- all mass and volume a point
start
to=0
do=0
end
t
d
time (t) : use to keep track of the object
at a particular instant - synchronize stop watches
time interval or an instant in time
start at t=0
to
Position (d) : rectilinear - displacement, distance
where the object is
start-position at t=0
do
stop watch at end
time t, position d
Speed and Velocity
SPEED (scalar) – rate of change in position
how far in a given time – how fast
AVERAGE SPEED – OVER A TIME INTERVAL
distance
vAVG = time
=d/t
can speed up or slow down during trip
faster – cover more distance in a given time
what constant speed to cover a distance in a given time
INSTANTANEOUS SPEED – AT A PARTICULAR TIME
like looking at speedometer at an instant
v
measure: average speed in a short interval
Speed of sound – uniform motion (constant speed)
v = vAVG = 1100 ft/s
Light faster than sound
Hear thunder
t=5s
See lightning
to=0s
d
How far away is the lightning strike?
Velocity – how fast and what direction
VECTOR!
Magnitude – speed
Direction - which way it’s going
Rectilinear – speed is magnitude of velocity
direction – left/right or up/down
y
+
Direction using normal
coordinate directions
+ x
AVERAGE VELOCITY – time interval again
vavg
displacement
=
=d/t
time
INSTANTANEOUS VELOCITY
at each instant during the interval -- short piece of time interval
Speed and velocity
t = 10 min.
20 feet
Not same
for 2D
CHANGES IN VELOCITY
speeding up
slowing down
changing direction
}
acceleration – rate of
change in velocity
can feel – force (cars, elevators,..)
VECTOR
a
t
d
to=0
do=0
vo =
initial instantaneous
velocity
v=
Final instantaneous
velocity
rectilinear – change in speed
Average and instantaneous –
UNIFORM (CONSTANT) ACCELERATION
change in speed
a=
= (v-vo)
time
t
Two types of problems: Uniform motion
Uniform acceleration
Tools for 1D Uniform Acceleration - formulae
DEFINITIONS :
vavg = d / t
a = (v-vo)/ t
DERIVATIONS : not magic
– use math to rearrange definitions
UNIFORM ACCELERATION FORMULA
d=vot + ½ at2
v=vo + at
vavg= (v+vo)/2 vavg=d / t
can find out how objects move
Only works for constant acceleration.
Identifies when
acceleration begins
to=0
do=0
vo =
a
t
d
Fill out line diagram
and apply formula
v=
How to pick right formula
-pick two with variable of interest (v, t, a, d)
-find variable you don’t care about
1D uniform acceleration - examples
1. An airplane starts from rest and reaches the
final take-off velocity in 50 s. What is its acceleration?
2. Space shuttle rocket accelerates 2 m/s2. a) What is
the velocity 90 s after lift-off?
b) How far did it go?
Finish description
3. A car travels 75 ft/s east and slams on brakes. a) If it
stops in a distance of 200 ft, then how long did it take
to stop?
b) What is the acceleration?
4. Car accelerates on I-10 uniformly along ramp. The
speed is 30 ft/s at one instant and 5 s later it is
going 110 ft/s. What is the acceleration of the car?
2D Motion (planar) - projectiles
supported by
Unnatural motion to Aristotle
Church 2000 yrs
four elements
natural state – rest (he said so)
medium adds resistance to speed
slow / fast – depends on object
unnatural – medium pushes object
DEMO – weight vs. crumpled paper
ARISTOTLE RIGHT?
Galileo – scientific method
measure the motion - kinematics
compare quantitatively
inclined plane
all objects fall the same
rate: ay = - g = -10 m/s2
( ay = - g
g = 10 m/s2)
natural state – constant velocity
everything we need to treat complex 2D motion
projectile motion :objects projected
Planar Motion
two dimensional motion
in the plane of paper
x and y graph
describe vector position - vector velocity
EXAMPLES
Uniform Circular Motion:
ball on a string
Moves at a constant
speed in circle
y
, x
At time t :
position given by x and y
Projectile Motion:
cannonball, arrow
vo
Projectile:
- projected (thrown) with initial velocity
- falls in earth’s gravity
Projectile motion
thrown in the earth’s gravity
separate into component directions (x,y)
work with each component separately!
VERTICAL DIRECTION
Galileo:
ALL objects fall the same
ay= -g
g =10 m/s2 approx.
ay= - g =-10 m/s2
to=0
yo=0
voy =
t=
y=
vy =
HORIZONTAL DIRECTION
No acceleration if no force (friction)
ax=0 (ignore air resistance)
ax=0
to=0
xo=0
vox =
t=
x=
vx =
Two Cases We Will Deal With:
Vertical Projectile - Jump ball
FREE FALL - freely falling in gravity
thrown straight up-fall straight down
rectilinear - y direction
ay
to=0
yo=0
voy =
t=
y=
vy =
a) time to highest point?
b) how high?
EXAMPLES
A baseball is thrown straight up with a
speed of 35 m/s. a)How long does it take to reach
the highest point? b) How high does the ball go?
An egg is thrown straight downward with a
speed of 12 m/s. What is its speed 3 seconds later?
Horizontal projectile - cannonball, soccer, gun
Projected with a horizontal velocity
-no initial velocity in vertical (y) direction
-must work in two directions - separate !
-time connects the directions
Separate components
vertical
free fall
horizontal uniform motion
two rectilinear problems
12 m
vo=40 m/s
range
a) How much time to hit ground?
what direction?
to=0
yo=0
voy =
t=
y=
vy =
B) What is the range of the projectile?
direction?
time connects position for fall in both directions:
how far does it go in x-direction in
the time it takes to fall?
to=0
xo=0
vox =
t=
x=
vx =
NOTE: time of fall has nothing to
do with x-direction!
falls same independent of horizontal speed!!!!
Motion - Part 2:
Dynamics (MECHANICS)-Why objects move!
not unnatural motion
* force not required to keep object going
* well-defined laws of motion
Sir Isaac Newton - 1st theoretical physicist
Great mathematician
bubonic plague sent him home to orchard
developed theories in 18 monthsalgebra, calculus, motion, gravitation
fluid motion, optics (Principia, 1687)
looked at results of others---Galileo, Keppler
“on the shoulders of great men”
Newton’s Laws explain events in everyday life!
Newton’s First Law of Motion
A body in uniform motion will remain in uniform
motion unless acted on by an external force
new natural state - uniform motion
change motion with force - acceleration
so objects travel in a straight line at a constant
speed unless force (push or pull) acts
– natural state
inertia - tendency of an object to remain
in uniform motion
LAW OF INERTIA
Decelleration – turning a curve
PROJECTILE MOTION
Galileo knew – but Newton published
Newton’s Second Law of Motion (Force Law)
The acceleration of a body is proportional to the
force and inversely proportional to the mass
a=F/m
m proportionality constant
(inertial) mass (kg) – resistance to a change
in uniform motion-or force
-ability to remain in uniform motion
big mass – small acceleration
small mass – accelerates easily
train vs. bicycle
Example:
How much force is required to accelerate
a 1 kg block at 1 m/s2?
MODEL
a=1 m/s2
F=?
m=1 kg
a=F/m
F=ma
F= ma = (1 kg)(1 m/s2) = 1 kg m/s2
SI force: kg m/s2 = 1 Newton = 1 N
F=ma not whole truth
F=ma
vector equation – direction
2N left gives 2 m/s2 left
Also talk about
net force – add all forces on object
Fnet=Fpush + (-Ffriction)
friction opposes motion
Second Law Examples:
m=1000 kg
1.
a=1 m/s2
F=?
2. two people
m=1000 kg
F1=500 N
a=?
F2=800 N
Fnet =
3.
F1=500 N
m=1000 kg
a=?
F2= 800 N
Direction – be careful!!!
Weight and mass
Mass – intrinsic property of matter
- doesn’t change
-always resistance to force
Weight – force of gravity on an object
- depends on location
-difficulty in lifting an object
-weightless in space, but F=ma still
no effort to lift!
weight – force of gravity
W= F = ma = mg on Earth surface
g=10 m/s2 acc. due to gravity
Surface gravity : depends on gsurface
Moon
gmoon=1/6 gearth
=1.66 m/s2
M=100 kg
Wearth= 1000 N
Wmoon= 166 N
Wspace= 0
g=0 weightless
F=ma still
Newton’s laws good for more than 200 years
NO DISCREPANCIES
Tribute to Newton
20th centurynew observations
measurement techniques improved
failures in F=ma
Jet planes, rockets – very very fast scale
Einstein’s Theory of Special Relativity
E-Microscope, scattering – very very small scale
Quantum Mechanical Theory (Bohr, Scrodinger)
Telescopy (BH, Neutron stars) – very very massive scale
Einstein’s Theory of General Relativity
ALL theories reduce to F=ma in the scale
of everyday experience:
- use Newton’s 2nd law for our purposes
- F=ma valid for cars, buildings, etc.
less complicated math!!!!
- use other theories when needed….
Forces of Nature: accelerate objects
Gravitational – force between masses
suns, planets, people
Electromagnetic – force between charges
opposites attract-likes repel
“contact force”
Strong Nuclear – nucleus of an atom
keeps atom together
“ likes repel”
Weak Nuclear – nuclear decay
gamma rays, beta-decay, nuclear reactions
UNIFYING THEORY –
binds all forces together at times beginning
-all forces have same form
HOLY GRAIL
Newton’s Third Law of Motion (Action-Reaction)
If one body exerts a force on a second, then
the second exerts a force back on the first
which is equal in magnitude and opposite
in direction
Easy statement:
How objects push on one another
- Forces exerted in pairs
-Always exerts force back
-Large mass, small acceleration
pendulum toy
leaning on wall
walking
Fhand
Fwall
Application of Newton’s Laws: Circular Motion
Uniform Circular motion object traveling in a
circle of constant radius at uniform speed
Like planet motion
Or a ball on a string
1st law – inertial movement
constant speed in
straight line - velocity
R
Velocity – tangent
Acceleration - radial
2nd law – forced
falls in toward center
– direction change - acceleration
Centripetal (center-seeking) acceleration - FORCE
acceleration required to keep object on circle
too fast, spirals in – too slow, spirals out
ac=v2 / R
depends on particular circle and speed
Another Application: MOMENTUM
momentum – difficulty in stopping an object
p = mv linear momentum
mass and velocity
vector – direction!
BASEBALL
m=0.2 kg
v=40 m/s
p=mv=(0.2 kg)(40 m/s) = 8 kg m/s SI units
kg m/s is almost N
TRAIN
m=100,000 kg
v=1 m/s
p=mv=(100,000 kg)(1 m/s) = 100,000 kg m/s
Heavy or moving fast
harder to stop!
no motion—no momentum
You can change the motion by
changing momentum
accelerate-fell the force from
momentum changes
fastball hurts more than slider
DERIVATION: Impulse
Second law
definition:acceleration
Fext=ma
a= (v-vo)/ t
F=ma=m{(v-vo)/ t}
IMPULSE-MOMENTUM THEOREM
I = Ftc = mv-mvo
Impulsechange in momentum
tc contact time -during which force applied
external force – accelerates as long as force applied
Consequences:
sports – tennis
golf
baseball
Hit ball as hard as possible
and
Follow-through (increase tc)
}
safety (cars) -- metal dash
padded dash
Io=Ftc
airbags
}
SAME IMPULSE-
increase
tc
IMPULSE – force applied for a time
external force produces acceleration
accelerates for time tc
a
to=0
xo=0
vox =
t =tc
x=
vx =
Impulse momentum theorem includes acceleration
I= F tc = mv-mvo
EXAMPLE:
A baseball is initially pitched toward the batter
at 40 m/s, and the batter hits it straight back to
the pitcher at 30 m/s.
a) What impulse is imparted to the ball?
b) What is the force on the bat?
The bat applies the external force which
changes the motion of the ball
external
– connected to body
– connected to ground - etc
Conservation of Momentum - COLLISIONS
Momentum important in collisions
COLLISION MODEL
v2
v1
BEFORE
m2
m1
Isolated
System
No external
forces
individual impulses cancel –equal & opposite
F21
F12
DURING
accelerated
m1 m2 internal forces
3rd lawaction-reaction pair
v2
v1
AFTER
m1
m2
Momentum exchanged : I is change in momentum
I12=-I21
one gains, other loses momentum
Conservation laws
conserved – same before as after
constant if assumptions true
Conservation of momentum
ptot=m1v1+m2v2
if no external forces
INTERACTING OBJECTS
For collisions, conserved before and after collision
ptot= (m1v1+m2v2)before =(m1v1+m2v2)after
Internal forces transfer momentum
ISOLATED FROM OUTSIDE FORCES
No momentum lost
transferred
EXAMPLE:
perfectly inelastic – stick together after colliding
Newton’s Law of Universal Gravitation
Newton described how a gravitational force would act
MOTIVATION:
ASTRONOMY – circular motion
inertial- - linear
centripetal
Falls toward center of earth
MOON
R
Moon falls like apple
Earth
What caused
moon to fall?
APPLES and the MOON
fall due to same force -- gravity
LAW OF UNIVERSAL GRAVITATION
All objects with mass attract all other
objects with mass
-attractive force
-smallest force in nature
-universal (all objects the same)
falling apples-orbiting planets-satellites
EXPLAINED Heliocentric Model!
Gravitation Model – picture to understand
d
point mass-centers
m1
m2
F=G m1m2 / d2
m 1, m 2
d
G
masses (kg)
separation (m)
universal constant
same for everything
G
must be measured
Newton couldn’t do that, but he could:
1. explain motions of planets around Sun
(satellites, comets)
2. explain the tides from moon
3. explain why g changes w/ altitude
(distance from center of earth)
4. orbital perturbations – deviations
from predicted path
Cavendish experiment
Established the universal gravitation constant
G
G = 6.67 x 10-11 N m2/kg2
Can do things like:
- calculate forces between ordinary objects
- ’weigh’ the earth
- predict new planets (perturbations)
- put man-made satellites into orbit
centripetal force equals gravity force
TV, mapping, weather, spy
geosynchronous – same period as earth