ap physics 1 chapter 5 circular motion, newton’s universal law of gravity, and kepler’s laws
Post on 19-Jan-2016
228 Views
Preview:
TRANSCRIPT
AP Physics 1 Chapter 5Circular Motion, Newton’s Universal Law of
Gravity, and Kepler’s Laws
Centripetal force at work!
Hammer throw
Tangential Velocity and Centripetal Acceleration
Roadway banking
Sitges Terramar in Spain(60 degree bank)
Roadway Banking at an extreme
Speed/Velocity in a Circle
T
r
T
dscircle
2
Consider an object moving in a circle around a specific origin. The DISTANCE the object covers in ONE REVOLUTION is called the CIRCUMFERENCE. The TIME that it takes to cover this distance is called the PERIOD.
Speed is the MAGNITUDE of the velocity. And while the speed may be constant, the VELOCITY is NOT. Since velocity is a vector with BOTH magnitude AND direction, we see that the direction o the velocity is ALWAYS changing.
We call this velocity, TANGENTIAL velocity as its direction is draw TANGENT to the circle.
Centripetal Acceleration
metersin length arc
sr
s
v
v
r
vt
vtsv
v
r
s
onacceleratilcentripetaa
at
v
r
v
c
c
2
Suppose we had a circle with angle, between 2 radaii. You may recall:
vo
v
v
vov
Centripetal means “center seeking” so that means that the acceleration points towards the CENTER of the circle
Drawing the Directions correctly
r
va
T
rv cc
22
So for an object traveling in a counter-clockwise path. The velocity would be drawn TANGENT to the circle and the acceleration would be drawn TOWARDS the CENTER.
To find the MAGNITUDES of each we have:
Circular Motion and N.S.L
Recall that according to Newton’s Second Law, the acceleration is directly proportional to the Force. If this is true: ForcelCentripetaF
r
mvFF
r
vamaF
c
cNET
cNET
2
2
Since the acceleration and the force are directly related, the force must ALSO point towards the center. This is called CENTRIPETAL FORCE.
NOTE: The centripetal force is a NET FORCE. It could be represented by one or more forces. So NEVER draw it in an F.B.D.
Examples
T
rvc
2 smvc /26.4
)4*28(.
)76(.2
222
/92.2376.0
)26.4(sm
r
vac
The blade of a windshield wiper moves through an angle of 90 degrees in 0.28seconds. The tip of the blade moves on the arc of a circle that has a radius of 0.76m. What is the magnitude of the centripetal acceleration of the tip of the blade?
Examples
rg
v
r
mvmg
r
mvF
FF
N
cf
2
2
2
187.0)8.9)(15.0(
)524.0(
/524.080.1
)15.0(22
sec80.1555.0
sec1
sec555.0sec60
min1*
min3.33
22
rg
v
smT
rv
Trevrev
revrev
c
What is the minimum coefficient of static friction necessary to allow a penny to rotate along a 33 1/3 rpm record (diameter= 0.300 m), whenthe penny is placed at the outer edge of the record?
mg
FN
Ff
Top view
Side view
ExamplesVenus rotates slowly about its axis,
the period being 243 days. The mass of Venus is 4.87 x 1024 kg. Determine the radius for a synchronous satellite in orbit around Venus. (assume circular orbit)
3
2
272411
32
2
2
23
2
22
2
2
2
4
)101.2)(1087.4)(1067.6(
44
4
2
xxxr
GMTr
GMTr
T
r
r
GM
T
rvv
r
GMr
mv
r
MmGFF
c
cg
Fg
1.54x109 m
Examples
The maximum tension that a 0.50 m string can tolerate is 14 N. A 0.25-kg ball attached to this string is being whirled in a vertical circle. What is the maximum speed the ball can have (a) the top of the circle, (b)at the bottom of the circle?
smvm
mgTrv
mvmgTrr
mvmgT
r
mvmaFF ccNET
/74.525.0
))8.9)(25.0(14(5.0)(
)( 22
2
mgT
Examples
smvm
mgTrv
mvmgTrr
mvmgT
r
mvmaFF ccNET
/81.425.0
))8.9)(25.0(14(5.0)(
)( 22
2
At the bottom?
mg
T
Newton’s Law of GravitationWhat causes YOU to be pulled down? THE EARTH….or
more specifically…the EARTH’S MASS. Anything that has MASS has a gravitational pull towards it.
MmFgWhat the proportionality above is saying is that for there to be a FORCE DUE TO GRAVITY on something there must be at least 2 masses involved, where one is larger than the other.
N.L.o.G.As you move AWAY from the earth, your DISTANCE increases and your FORCE DUE TO GRAVITY decrease. This is a special INVERSE relationship called an Inverse-Square.
2
1
rFg
The “r” stands for SEPARATION DISTANCE and is the distance between the CENTERS OF MASS of the 2 objects. We us the symbol “r” as it symbolizes the radius. Gravitation is closely related to circular motion as you will discover later.
N.L.o.G – Putting it all together
221
2
227
221
1067.6
Constant nalGravitatio UniversalG
alityproportion ofconstant
r
mmGF
kgNmxG
Gr
mmF
g
g
earth eLEAVING th areyou when thisUse
earth on the areyou when thisUse
221
r
mmGF
mgF
g
g
Try this!
earth eLEAVING th areyou when thisUse
earth on the areyou when thisUse
221
r
mmGF
mgF
g
g
mxr
kgxM
r
MGg
r
MmGmg
6
24
2
2
1037.6 Earth theof radius
1097.5Earth theof Mass
Let’s set the 2 equations equal to each other since they BOTH represent your weight or force due to gravity
SOLVE FOR g!
226
2427
/81.9)1037.6(
)1097.5)(1067.6(sm
x
xxg
Kepler’s Laws
Testing Models
Geocentric (or Ptolemaic) means the Earth is at the center and motionless.
Heliocentric (or Copernican) means the Sun is at the center and motionless.
Scholars wanted to differentiate models by comparing the predictions with precise observations.
This originated the modern scientific method.
Kepler’s Work
Tycho Brahe led a team which collected data on the position of the planets (1580-1600 with no telescopes).
Mathematician Johannes Kepler was hired by Brahe to analyze the data.
He took 20 years of data on position and relative distance.
No calculus, no graph paper, no log tables.
Both Ptolemy and Copernicus were wrong.
He determined 3 laws of planetary motion (1600-1630).
Kepler’s First Law
The orbit of a planet is an ellipse with the sun at one focus.
A path connecting the two foci to the ellipse always has the same length.
Orbital Description
An ellipse is described by two axes. Long – semimajor (a) Short – semiminor (b)
The area is ab (becomes r2 for a circle).
b
a
Orbital Speed
The centripetal force is due to gravity. GMm/r2 = mv2/r v2 = GM/r
Larger radius orbit means slower speed.
Within an ellipse larger distance also gives slower speed.
Kepler’s Second Law
The line joining a planet and the sun sweeps equal areas in equal time.
The planet moves slowly here.
The planet moves quickly here.
t
t
Orbital Period
The speed is related to the period in a circular orbit. v2 = GM/r (2r/T)2 = GM/r T2 = 42r3/GM
Larger radius orbit means longer period.
Within an ellipse, a larger semimajor axis also gives a longer period.
Kepler’s Third Law
The square of a planet’s period is proportional to the cube of the length of the orbit’s semimajor axis. T2/a3 = constant The constant is the same for all objects orbiting
the Sun
semimajor axis: a
direction of orbit
The time for one orbit is one period: T
Hyperbolic Orbits
Some satellites have so much speed that gravity can’t hold them in an orbit.
These objects take a hyperbolic orbit that never returns.
top related