### Chapter 8: Motion in Circles

```CPO Science
Foundations of Physics
Chapter 9
Unit 3, Chapter 8
Unit 3: Motion and Forces in 2 and 3
Dimensions
Chapter 8 Using Vectors: Forces and Motion
 8.1 Motion in Circles
 8.2 Centripetal Force
 8.3 Universal Gravitation and Orbital
Motion
Chapter 8 Objectives
1. Calculate angular speed in radians per second.
2. Calculate linear speed from angular speed and viceversa.
3. Describe and calculate centripetal forces and
accelerations.
4. Describe the relationship between the force of gravity
and the masses and distance between objects.
5. Calculate the force of gravity when given masses and
distance between two objects.
6. Describe why satellites remain in orbit around a
planet.
Chapter 8 Vocabulary Terms
 rotate
 revolve
 orbit
 axis
 law of universal
gravitation
 centripetal force
 centripetal
acceleration
 circumference
 linear speed
 ellipse
 satellite
 angular speed
 centrifugal force
 angular displacement
 gravitational constant
8.1 Vectors and Direction
Key Question:
How do we describe circular motion?
*Students read Section 8.1 AFTER Investigation 8.1
8.1 Motion in Circles
 We say an object rotates
about its axis when the
axis is part of the moving
object.
 A child revolves on a
merry-go-round because
he is external to the merrygo-round's axis.
8.1 Angular Speed
 Angular speed is the rate at
which an object rotates or
revolves.
 There are two ways to measure
angular speed
— number of turns per unit of time
(rotations/minute)
— change in angle per unit of time
8.1 Angular Speed
 For the purpose of
angular speed, the
radian is a better unit
for angles.
 One radian is approx.
57.3 degrees.
 Radians are better for angular speed because a
radian is a ratio of two lengths.
8.1 Angular Speed
Angular speed
w=q
t
Time taken (sec)
8.1 Calculate angular speed
 A bicycle wheel makes
six turns in 2 seconds.
 What is its angular speed
in radians per second?
8.1 Linear and Angular Speed
 A wheel rolling along the ground has both a linear speed
and an angular speed.
 A point at the edge of a wheel moves one circumference
in each turn of the circle.
8.1 Linear and Angular Speed
Circumference
(m)
C=2Pr
Distance (m)
Speed
(m/sec)
v = d2 P r
t
Time (sec)
8.1 Linear and Angular Speed
Linear speed
(m/sec)
v=wr
Angular speed
*This formula is used in automobile
speedometers based on a tire's radius.
8.1 Calculate linear from angular speed
 Two children are spinning
around on a merry-go-round.
 Siv is standing 4 meters from the axis of rotation
and Holly is standing 2 meters from the axis.
 Calculate each child’s linear speed when the
angular speed of the merry go-round is 1 rad/sec.
8.1 Linear and Angular Speed and
Displacement
8.1 Calculate angular from linear speed
 A bicycle has wheels that are
70 cm in diameter (35 cm
 The bicycle is moving forward with a linear speed
of 11 m/sec.
 Assume the bicycle wheels are not slipping and
calculate the angular speed of the wheels in RPM.
8.2 Centripetal Force
Key Question:
Why does a roller coaster stay on a track upside
down on a loop?
*Students read Section 8.2 AFTER Investigation 8.2
8.2 Centripetal Force
 We usually think of acceleration as a change in speed.
 Because velocity includes both speed and direction,
acceleration can also be a change in the direction of
motion.
8.2 Centripetal Force
 Any force that causes an object to move in a circle is
called a centripetal force.
 A centripetal force is always perpendicular to an
object’s motion, toward the center of the circle.
8.2 Centripetal Force
Mass (kg)
Centripetal
force (N)
Fc = mv2
r
Linear speed
(m/sec)
(m)
8.2 Calculate centripetal force
 A 50-kilogram passenger on an amusement park ride
stands with his back against the wall of a cylindrical
room with radius of 3 m.
 What is the centripetal force of the wall pressing into
his back when the room spins and he is moving at 6
m/sec?
8.2 Centripetal Acceleration
 Acceleration is the rate at which an object’s velocity
changes as the result of a force.
 Centripetal acceleration is the acceleration of an
object moving in a circle due to the centripetal force.
8.2 Centripetal Acceleration
Centripetal
acceleration (m/sec2)
ac = v2
r
Speed
(m/sec)
(m)
8.2 Calculate centripetal acceleration
 A motorcycle drives around a bend with a 50-meter
radius at 10 m/sec.
 Find the motor cycle’s centripetal acceleration and
compare it with g, the acceleration of gravity.
8.2 Centrifugal Force
 We call an object’s tendency to
resist a change in its motion its
inertia.
 An object moving in a circle is
constantly changing its direction
of motion.
 Although the centripetal force pushes you toward the
center of the circular path...
 ...it seems as if there also is a force pushing you to the
outside. This apparent outward force is called
centrifugal force.
8.2 Centrifugal Force
 Centrifugal force is not a true
force exerted on your body.
 It is simply your tendency to
move in a straight line due to
inertia.
 This is easy to observe by twirling a small object at the
end of a string.
 When the string is released, the object flies off in a
straight line tangent to the circle.
8.3 Universal Gravitation and Orbital
Motion
Key Question:
How strong is gravity in other places in the
universe?
*Students read Section 8.3 AFTER Investigation 8.3
8.3 Universal Gravitation
and Orbital Motion
 Sir Isaac Newton first deduced that
the force responsible for making
objects fall on Earth is the same
force that keeps the moon in orbit.
 This idea is known as the law of
universal gravitation.
 Gravitational force exists between
all objects that have mass.
 The strength of the gravitational
force depends on the mass of the
objects and the distance between
them.
8.3 Law of Universal Gravitation
Force (N)
F = m1m2
r2
Mass 1
Mass 2
Distance between
masses (m)
8.3 Calculate gravitational force
 The mass of the moon is
7.36 × 1022 kg.
 The radius of the moon is
1.74 × 106 m.
 Use the equation of universal gravitation to calculate
the weight of a 90 kg astronaut on the surface of the
moon.
8.3 Orbital Motion
 A satellite is an object that is
bound by gravity to another
object such as a planet or
star.
 If a satellite is launched above
Earth at more than 8
kilometers per second, the
orbit will be a noncircular
ellipse.
 A satellite in an elliptical orbit
does not move at a constant
speed.
Application: Satellite Motion
```