Transcript Circular Motion and Gravitation

Chapter 7: Rotational Motion
and the Law of Gravity
Objectives
• Be able to distinguish between a rotation and
a revolution.
• Be able to distinguish between frequency and
period.
• Be able to calculate tangential speed.
• Understand the concept of a centripetal
acceleration.
Circular Motion
revolution: object moving
in a circular (or elliptical)
path around an axis point
rotation: object spinning
around its axis
period (T): time required for one complete cycle
frequency (f): number of cycles per unit time T  1
f
hertz (hz): cycles/second
Uniform Circular Motion
v
tangential speed
d 2  r
v 
 2  r  f
t
T
r
What is the tangential speed (in m/s)
of a palm tree on the equator? What
is it for a Ponderosa pine in Polson?
Rearth = 6380 km
Centripetal Acceleration
Dv = vf – vi = vf + (– vi)
vi
r
q
vf
d
vf
Dv
q
– vi
q
r
centripetal acceleration (ac): a
center-seeking change in velocity
Dv d

v
r
Dv v  t

v
r
Dv v 2

t
r
v2
ac 
r
Objectives
• Understand the concept of centripetal force.
• Be able to identify or give examples of forces
acting as centripetal forces.
• Be able to solve centripetal force problems.
Centripetal Force
centripetal force:
any center-seeking
force that results
in circular motion
v
Fc
v
v
Fc is unbalanced: it causes a
change in velocity. Fc and v are
perpendicular: no net work is
done by Fc so the KE (and speed)
remains constant.
F  m  a
Fc  m  a c
m v2
Fc 
r
m  ( 2  r T ) 2
Fc 
r
m  4 2  r
Fc 
2
T
Centripetal Forces
Forces acting as centripetal forces:
hammer throw
(tension)
motorcycle cage
(normal force)
car turning on road
(friction)
moon orbiting earth (gravity)
e- orbiting nucleus
(electromagnetic)
Centripetal Force
At what maximum speed that a car make a turn of
radius 12.3 meters if the coefficient of friction
between the tires and the road is 1.94?
What is the magnitude of the Fc if the mass of the
car is 1383 kg?
Twirl-O Problem
On the popular Twirl-O, a passenger is held inside a
large spinning cylinder. If the radius of the ride is
4.0 m, with what rotational period must the ride
rotate in order for the passenger to not fall? The ms
between the wall and the passenger is 0.60.
Objectives
• Understand how Newton’s third law relates to
the concept of a “centrifugal” force.
• Explain how simulated gravity could be
achieved on a spacecraft.
• Be able to solve simulated gravity problems.
“Centrifugal Force”
The force equal-and-opposite to a centripetal force
is known as a centrifugal force.
can on bug (FC is FN)
bug on can (~ FW)
From the bug’s point of view, it feels like the
normal force exerted upward by the ground.
Simulated Gravity
FN = FC
FW
simulated weight (FW) = FC = m · aC = m · g
4 r
aC  2
T
2
A simulated gravity can be produced
by adjusting r and T.
If r = 95 m, what does T need to be ?
Centripetal Force Extra-Credit
At what minimum height will a Hot Wheels car make it
around the loop-the-loop without falling? Hint: at the
top of the loop the only force acting is Fw (= Fc)
h=?
find the
equation
r
Objectives
• Explain the factors that affect the force of
gravity between two objects.
• Understand the concept of the universal
gravitational constant, G.
• Be able to solve gravitation problems.
Universal Gravitation
1660s: Isaac Newton first realized
that gravity keeps the moon in
orbit around the earth (FG = Fc)
gravity: an attractive force between two masses
What factors affect the strength of the force?
FG ~ m1· m2
FG ~ 1 / r2
m1  m2
FG ~
r2
m1  m2
FG  G 
r2
Universal Gravitational Constant
• “Big G” was first measured by Cavendish in 1797
• G = 6.67 x 10-11 Nm2/kg2
Mass of the Earth
The earth has a radius of 6380 km. If a 1.0 kg mass
weighs 9.81 N, what is the mass of the earth?
Universal Gravitation Problem
How much gravitational force does the sun (150
million km away = 1 AU) exert on a 65 kg person?
Msun = 2.00 x 1030 kg.
Objectives
• Be familiar with Kepler’s third law.
• Understand how his law can be derived.
• Perform calculations related to the law.
A Brief History of Astromony
• Ptolemy, Aristotle, and the
Catholic Church: geocentric
model
• Aristarchus, Aryabhata,
Copernicus: heliocentric
model
• Galileo: moons orbit Jupiter
• Kepler develops 3 laws of
orbital motion
Kepler’s Third Law
• Johannes Kepler (1619):
r3/T2 = 1 for all planets
in our solar system
• r = # AU and T = # yrs
Planet
T
(yrs)
r
(AU)
T2
r3
Mercury
0.24
0.39
0.06
0.06
Venus
0.62
0.72
0.39
0.37
Earth
1.00
1.00
1.00
1.00
Mars
1.88
1.52
3.53
3.51
Jupiter
11.9
5.20
142
141
Saturn
29.5
9.54
870
868
Kepler’s 3rd Law Proof
FG  FC
G
mp  M S
r
2

m p  4 2 r
T2
G  MS
r3
 2
2
4
T
3
3
rA
rB
For any pair of satellites

2
2
TA
TB
orbiting the same star/planet.
What is the orbital period of Jupiter if r = 5.2 AU?
Objectives
• Be able to explain why the same side of the
moon always faces the earth.
• Be able to explain how the force of gravity
relates to ocean tides.
• Understand the concept of a black hole.
The Moon’s Orbit
• center of mass ≠ center of gravity
• as it orbits, the same side of moon must face the
earth
• rotational T = orbital T
The Tides
• FG ~ 1/r2, so FA > FB > FC,
• tidal bulges form (not to scale!)
• two high, two low tides daily (polar view)
Tidal Forces
• Fg of the sun is 180 X greater than the moon
• but Fg from moon has 2X greater difference:
SUN on EARTH
Near side: 3.5456 x1022 N
Far side:
3.5452 x1022 N
Difference: 0.0004 x1022 N
MOON on EARTH
Near side: 0.0207 x1022 N
Far side:
0.0198 x1022 N
Difference: 0.0009 x1022 N
Twice as much!
Tides
quarter
moons
new moon
(most extreme)
full moon
Extreme Tides
The tides are
most extreme
(higher and lower)
at higher latitudes
15 m at Bay of Fundy, Nova Scotia