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Chapter 7
Rotational Motion
Universal Law of Gravitation
Kepler’s Laws
Angular Displacement
- (^) Circular motion about AXIS
- (^) Three measures of angles:
- Degrees
- Revolutions (1 rev. = 360 deg.)
- Radians (2! rad.s = 360 deg.)
Angular Displacement, cont.
distance of a point:
s = 2! r N (N counts revolutions)
= r " ( " is in radians)
Example 7.
An automobile wheel has a radius of 42 cm. If a
car drives 10 km, through what angle has the
wheel rotated?
a) In revolutions
b) In radians
c) In degrees
a) N = 3789
b) " = 2.38x
4 radians
c) " = 1.36x
6 degrees
Angular Speed • Can be given in
- (^) Revolutions/s
- Radians/s --> Called #
- Linear Speed at r
" f # " i
t
in radians
v = 2! r "
N revolutions
t
2! r
# f $ # i (in rad.s)
t
v =! r
Example 7.
A race car engine can turn at a maximum rate of 12,
rpm. (revolutions per minute).
a) What is the angular velocity in radians per second.
b) If helipcopter blades were attached to the
crankshaft while it turns with this angular velocity, what
is the maximum radius of a blade such that the speed of
the blade tips stays below the speed of sound.
DATA: The speed of sound is 343 m/s a) 1256 rad/s
b) 27 cm
Angular Acceleration
- (^) Denoted by $
- (^) # must be in radians per sec.
- Units are rad/s!
- Every point on rigid object has same # and $
f
i
t
Rotational/Linear Equivalence:
! " #! x
# v
f
# v
f
% # a
t # t
Linear and Rotational Motion Analogies
Rotational Motion Linear Motion
(^ # 0 +^ #^ f )
t
! " = # 0 t +
$ t
2
! f =! 0 + " t
f
2
2
f
t $
% t
2
! x =
( v 0 +^ v^ f )
t
v f = v 0 + at
! x = v 0 t +
at
2
! x = v
f
t "
at
2
v
f
2
v 0
2
+ a! x
Example 7.
A pottery wheel is accelerated uniformly from rest
to a rate of 10 rpm in 30 seconds.
a.) What was the angular acceleration? (in rad/s
2 )
b.) How many revolutions did the wheel undergo
during that time?
a) 0.0349 rad/s
2
b) 2.50 revolutions
Linear movement of a rotating point
- Distance
- Speed
- Acceleration
Only works for angles in radians!
Different points have
different linear speeds!
x = r! "
v = r!
a = r!
Example 7.
A coin of radius 1.5 cm is initially rolling with a
rotational speed of 3.0 radians per second, and
comes to a rest after experiencing a slowing down of
$ = 0.05 rad/s
2 .
a.) Over what angle (in radians) did the coin rotate?
b.) What linear distance did the coin move?
a) 90 rad
b) 135 cm
Example 7.5c
a) Vector A
b) Vector B
c) Vector C
A
B
An astronaut is in C
cirular orbit
around the Earth.
Which vector might
describe the
gravitional force
acting on the
astronaut?
Example 7.6a
a) Vector A
b) Vector B
c) Vector C
A
B
C
Dale Earnhart drives
150 mph around a
circular track at
constant speed.
Neglecting air
resistance, which
vector best
describes the
frictional
force exerted on the
tires from contact
with the pavement?
Example 7.6b
a) Vector A
b) Vector B
c) Vector C
A
B
C
Dale Earnhart
drives 150 mph
around a circular
track at constant
speed.
Which vector best
describes the
frictional force
Dale Earnhart
experiences from
the seat?
Ball-on-String Demo
Example 7.
A space-station is constructed like a barbell with
two 1000-kg compartments separated by 50
meters that spin in a circle (r=25 m). The
compartments spin once every 10 seconds.
a) What is the acceleration at the extreme end of
the compartment? Give answer in terms of “g”s.
b) If the two compartments are held together by a
cable, what is the tension in the cable?
a) 9.87 m/s
2 = 1.01 “g”s
b) 9870 N
DEMO: FLYING POKER CHIPS
Example 7.
A race car speeds around a circular track.
a) If the coefficient of friction with the tires is 1.1,
what is the maximum centripetal acceleration (in
“g”s) that the race car can experience?
b) What is the minimum circumference of the track
that would permit the race car to travel at 300 km/
hr?
a) 1.1 “g”s
b) 4.04 km (in real life curves are banked)
Example 7.
A curve with a radius of
curvature of 0.5 km on a
highway is banked at an
angle of 20°. If the
highway were frictionless,
at what speed could a car
drive without sliding off
the road?
42.3 m/s = 94.5 mph
(Skip) Example 7.
A
A yo-yo is spun in a circle as
shown. If the length of the
string is L = 35 cm and the
circular path is repeated 1.
times per second, at what
angle " (with respect to the
vertical) does the string bend?
" = 71.6 degrees
Example 7.11a
Which vector represents acceleration?
a) A b) E
c) F d) B
e) J
Which vector represents net force acting on car?
a) A b) E
c) F d) B
e) J
Example 7.11b Example 7.11c
If car moves at "design" speed, which vector represents
the force acting on car from contact with road
a) D b) E
c) G d) I
e) J
Example 7.
5.99x
24 kg
Given: In SI units, G = 6.67x
g=9.81 and the radius of Earth is
6.38 x
6 .
Find Earth’s mass:
Example 7.
Given: The mass of Jupiter is 1.73x
27 kg
and Period of Io’s orbit is 17 days
Find: Radius of Io’s orbit
r = 1.85x
9
m
Tycho Brahe (1546-1601)
- (^) Lost part of nose in a duel
• EXTREMELY ACCURATE
astronomical observations, nearly
10X improvement, corrected for
atmosphere
- Believed in Retrograde Motion
- (^) Hired Kepler to work as
mathematician
Uraniborg (on an island near Copenhagen)
Johannes Kepler
- (^) First to:
- (^) Explain planetary motion
- Investigate the formation of
pictures with a pin hole
camera;
- Explain the process of vision
by refraction within the eye
- Formulate eyeglass designed
for nearsightedness and
farsightedness;
- Explain the use of both eyes
for depth perception.
Johannes Kepler (1571-1630)
- (^) First to:
- explain the principles of how a telescope works
- discover and describe total internal reflection.
- (^) explain that tides are caused by the Moon.
- suggest that the Sun rotates about its axis
- derive the birth year of Christ, that is now
universally accepted.
- derive logarithms purely based on mathematics
- (^) He tried to use stellar parallax caused by the Earth's
orbit to measure the distance to the stars; the same
principle as depth perception. Today this branch of
research is called astrometry.
Isaac Newton (1642-1727)
- Invented Calculus
- Formulated the universal law of
gravitation
- (^) Showed how Kepler’s laws could
be derived from an inverse-
square-law force
- (^) Invented Wave Mechanics
- Numerous advances to
mathematics and geometry
Example 7.16a
Astronaut Bob stands atop the highest mountain of
planet Earth, which has radius R.
Astronaut Ted whizzes around in a circular orbit at
the same radius.
Astronaut Carol whizzes around in a circular orbit of
radius 3R.
Astronaut Alice is simply falling straight downward
and is at a radius R, but hasn’t hit the ground yet.
Bob
Alice
Ted
Carol
Which astronauts experience
weightlessness?
A.) All 4
B.) Ted and Carol
C.) Ted, Carol and Alice
Example 7.16b
Astronaut Bob stands atop the highest mountain of
planet Earth, which has radius R.
Astronaut Ted whizzes around in a circular orbit at
the same radius.
Astronaut Carol whizzes around in a circular orbit of
radius 3R.
Astronaut Alice is simply falling straight downward
and is at a radius R, but hasn’t hit the ground yet.
Bob
Alice
Ted
Carol
Assume each astronaut weighs
w=180 lbs on Earth.
The gravitational force acting on
Ted is
A.) w
B.) ZERO
Example 7.16c
Astronaut Bob stands atop the highest mountain of
planet Earth, which has radius R.
Astronaut Ted whizzes around in a circular orbit at
the same radius.
Astronaut Carol whizzes around in a circular orbit of
radius 3R.
Astronaut Alice is simply falling straight downward
and is at a radius R, but hasn’t hit the ground yet.
Bob
Alice
Ted
Carol
Assume each astronaut weighs
w=180 lbs on Earth.
The gravitational force acting on
Alice is
A.) w
B.) ZERO
Example 7.16d
Astronaut Bob stands atop the highest mountain of
planet Earth, which has radius R.
Astronaut Ted whizzes around in a circular orbit at
the same radius.
Astronaut Carol whizzes around in a circular orbit of
radius 3R.
Astronaut Alice is simply falling straight downward
and is at a radius R, but hasn’t hit the ground yet.
Bob
Alice
Ted
Carol
Assume each astronaut weighs
w=180 lbs on Earth.
The gravitational force acting on
Carol is (^) A.) w
B.) w/
C.) w/
D.) ZERO
Example 7.16e
Astronaut Bob stands atop the highest mountain of
planet Earth, which has radius R.
Astronaut Ted whizzes around in a circular orbit at
the same radius.
Astronaut Carol whizzes around in a circular orbit of
radius 3R.
Astronaut Alice is simply falling straight downward
and is at a radius R, but hasn’t hit the ground yet.
Bob
Alice
Ted
Carol
Which astronaut(s) undergo an
acceleration g=9.8 m/s
2 ?
A.) Alice
B.) Bob and Alice
C.) Alice and Ted
D.) Bob, Ted and Alice
E.) All four
Gravitational Potential Energy
Earth’s surface
- For arbitrary altitude
- (^) Zero reference level is
at r=%
PE =! G
Mm
r
Graphing PE vs. position
PE =! G
Mm
r
Example 7.
You wish to hurl a projectile from the surface of the
Earth (R e
= 6.38x
6 m) to an altitude of 20x
6 m
above the surface of the Earth. Ignore rotation of the
Earth and air resistance.
a) What initial velocity is required?
b) What velocity would be required in order for the
projectile to reach infinitely high? I.e., what is the
escape velocity?
c) (skip) How does the escape velocity compare to the
velocity required for a low earth orbit?
a) 9,736 m/s
b) 11,181 m/s
c) 7,906 m/s
