Ch 15: Gravitation and
Circular Motion
M Sittig
AP Physics B
Summer Course 2012
2012年AP物理B暑假班
Uniform Circular Motion
Centripetal Acceleration

Centripetal = “toward the center”
Centripetal acceleration
2
Centripetal
acceleration
(m/s2)
v
ac 
r
Radius (m)
Tangential
velocity (m/s)
Centripetal force
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
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Centripetal force is whatever force is
causing the centripetal acceleration.
This can be the net force, or some other
force: friction, gravity, tension, magnetic
force…
And a useful shortcut:
mv
Fc  mac 
r
2
Three examples
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Mass on a string
Car on a curve


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Friction
Banked
Satellite in orbit
Mass on a String
Practice Problem

A girl whirls a stone in a horizontal circle
2.1 m above the ground by means of a string
154 cm long. The string breaks, and the
stone flies of horizontally and strikes the
ground 11.3 m away. What was the
centripetal acceleration of the stone while it
was in circular motion?
Practice Problem

To study circular motion, two
students use the hand-held device
shown above, which consists of a rod
on which a spring scale is attached. A
polished glass tube attached at the
top serves as a guide for a light cord
attached the spring scale. A ball of
mass 0.200 kg is attached to the
other end of the cord. One student
swings the ball around at constant
speed in a horizontal circle with a
radius of 0.500 m. Assume friction
and air resistance are negligible.
Practice Problem


a. Explain how the students,
by using a timer and the
information given above, can
determine the speed of the ball
as it is revolving.
b. How much work is done by
the cord in one revolution?
Explain how you arrived at
your answer.
Practice Problem


c. The speed of the ball is
determined to be 3.7 m/s.
Assuming that the cord is
horizontal as it swings, calculate
the expected tension in the cord.
d. The actual tension in the cord
as measured by the spring scale is
5.8 N. What is the percent
difference between this measured
value of the tension and the value
calculated in part c. ?
Practice Problem
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e. The students find that, despite
their best efforts, they cannot swing
the ball so that the cord remains
exactly horizontal.
i. Draw a free body diagram
showing the forces acting on the ball
and identify the force that each
vector represents.
ii. Explain why it is not possible
for the ball to swing so that the cord
remains exactly horizontal.
iii. Calculate the angle that the
cord makes with the horizontal.
Car on a Curve
Practice Problem

A turn of radius 100 m is being designed for
a speed of 25 m/s. At what angle should the
turn be banked so that a car can make the
turn even under conditions of no friction?
Satellite in Orbit

First, examine Newton’s Law of Gravitation:
Newton’s Law of Gravitation
Universal
gravitational
constant
(m3/kg ·s2)
Mass (kg)
GM1M 2
FG  
2
r
Force of
gravity (m/s2)
Radius (m)
Satellite in Orbit



But I thought the force of gravity on an
object (weight) is F = mg???
Try calculating G·M1/r2 with the mass and
radius of the Earth…
So F = (G M1M2)/r2 = M2·g
Example Problem
Practice Problem

At what distance above the surface of the
Earth should a geostationary satellite orbit?
Practice Problem

A planet is a uniform sphere of mass M. The length
of the day on the planet (that is, the period of
rotation about its axis) is T. The linear speed of the
points on the equator of the planet is v. Based on
this information, what would be the reading (in
newtons) of the bathroom scale on the equator if a
person of mass m stands on it?
Gravitational Potential Energy

Not used often, but know it.
Gravitational Potential Energy
Universal
gravitational
constant
(m3/kg ·s2)
Mass (kg)
GM1M 2
PEG  
r
Potential
Energy (J)
Radius (m)
Gravitational Potential Energy
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Wait, isn’t PEg = mgh? Two answers:
No, that’s difference in PE.
Yes, remember that g = G·ME/RE2, so if we
take the center of the Earth to be PE = 0,
then PEg = PEG. Also, this new equation
works for any two bodies.
PEg  mgh  m(G  M E RE )h  m(G  M E RE ) RE
2
2