Transcript Notes - Haiku for Ignatius

CHAPTER 8: MOTION IN CIRCLES
• 8.1 Circular Motion
• 8.2 Centripetal Force
• 8.3 Universal Gravitation and Orbital Motion
CHAPTER 8 OBJECTIVES
•
•
•
•
•
•
Calculate angular speed in radians per second.
Calculate linear speed from angular speed and
vice-versa.
Describe and calculate centripetal forces and
accelerations.
Describe the relationship between the force of
gravity and the masses and distance between
objects.
Calculate the force of gravity when given masses
and distance between two objects.
Describe why satellites remain in orbit around a
planet.
CHAPTER 8 VOCABULARY
 angular displacement
 linear speed
 angular speed
 orbit
 axis
 radian
 centrifugal force
 centripetal acceleration
 centripetal force
 circumference
 ellipse
 gravitational constant
 law of universal gravitation
 revolve
 rotate
 satellite
8.1 MOTION IN CIRCLES
Investigation Key
Question:
How do we
describe circular
motion?
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 merrygo-round because he is
external to the merry-goround's axis.
8.1 MOTION IN CIRCLES
• Earth revolves around the
Sun once each year while it
rotates around its north-south
axis once each day.
8.1 MOTION IN CIRCLES
• 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 (deg/sec or
rad/sec)
8.1 CIRCULAR MOTION
• 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 THE RELATIONSHIP BETWEEN
LINEAR AND ANGULAR SPEED
• The circumference is the distance around a circle.
• The circumference depends on the radius of the
circle.
8.1 THE RELATIONSHIP BETWEEN
LINEAR AND ANGULAR SPEED
• The linear speed (v) of a point at the edge of a
turning circle is the circumference divided by the
time it takes to make one full turn.
• The linear speed of a point on a wheel depends on
the radius, r, which is the distance from the center of
rotation.
8.1 THE RELATIONSHIP BETWEEN
LINEAR AND ANGULAR SPEED
Circumference
(m)
C = 2π r
Radius (m)
Distance (m)
Speed
(m/sec)
v = d2π r
t
Time (sec)
8.1 THE RELATIONSHIP BETWEEN
LINEAR AND ANGULAR SPEED
Linear speed
(m/sec)
v=wr
Radius (m)
Angular speed
(rad/sec)
*Angular speed is represented with
a lowercase Greek omega (ω).
CALCULATE LINEAR FROM ANGULAR
SPEED
Two children are spinning around on a merry-goround. 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?
1.
2.
You are asked for the children’s linear speeds.
You are given the angular speed of the merry-go-round and radius
to each child.
3.
4.
Use v = ωr
Solve:
For Siv: v = (1 rad/s)(4 m) v = 4 m/s.


For Holly: v = (1 rad/s)(2 m) v = 2 m/s.
8.1 THE UNITS OF RADIANS PER
SECOND
• One radian is the angle you
get when you rotate the
radius of a circle a distance
on the circumference equal
to the length of the radius.
• One radian is approximately
57.3 degrees, so a radian is
a larger unit of angle
measure than a degree.
8.1 THE UNITS OF RADIANS PER SECOND
• Angular speed
naturally comes out in
units of radians per
second.
• For the purpose of
angular speed, the
radian is a better unit
for angles.
 Radians are better for angular speed because a radian is
a ratio of two lengths.
8.1 ANGULAR SPEED
Angular speed
(rad/sec)
w=q
t
Angle turned (rad)
Time taken (sec)
CALCULATING ANGULAR SPEED
IN RAD/S
A bicycle wheel makes six turns in
2 seconds. What is its angular speed in
radians per second?
1.
2.
3.
4.
You are asked for the angular speed.
You are given turns and time.
There are 2π radians in one full turn. Use: ω = θ ÷ t
Solve: ω = (6 × 2π) ÷ (2 s) = 18.8 rad/s
8.1 RELATING ANGULAR SPEED, LINEAR
SPEED AND DISPLACEMENT
• As a wheel rotates, the point touching the ground
passes around its circumference.
• When the wheel has turned one full rotation, it has
moved forward a distance equal to its circumference.
• Therefore, the linear speed of a wheel is its angular
speed multiplied by its radius.
CALCULATING ANGULAR SPEED
FROM LINEAR SPEED
A bicycle has wheels that are 70 cm in diameter (35
cm radius). The bicycle is moving forward with a
linear speed of 11 m/s. Assume the bicycle wheels
are not slipping and calculate the angular speed of
the wheels in rpm.
1.
2.
You are asked for the angular speed in rpm.
You are given the linear speed and radius of the
wheel.
3. Use: v = ωr, 1 rotation = 2π radians
4. Solve: ω = v ÷ r = (11 m/s) ÷ (0.35 m) = 31.4 rad/s.
 Convert to rpm: 31.4 rad x 60 s x 1 rotation = 300 rpm
1s
1 min 2 π rad
CHAPTER 8: MOTION IN CIRCLES
• 8.1 Circular Motion
• 8.2 Centripetal Force
• 8.3 Universal Gravitation and Orbital Motion
INV 8.2 CENTRIPETAL FORCE
Investigation Key Question:
Why does a roller coaster stay on a track upside down
on a loop?
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 CALCULATING CENTRIPETAL FORCE
• The magnitude of the centripetal force needed to
move an object in a circle depends on the object’s
mass and speed, and on the radius of the circle.
8.2 CENTRIPETAL FORCE
Mass (kg)
Centripetal
force (N)
Fc = mv2
r
Linear speed
(m/sec)
Radius of path
(m)
CALCULATING 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?
1.
2.
3.
4.
You are asked to find the centripetal force.
You are given the radius, mass, and linear speed.
Use: Fc = mv2 ÷ r
Solve: Fc = (50 kg)(6 m/s)2 ÷ (3 m) = 600 N
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 = v 2
r
Speed
(m/sec)
Radius of path
(m)
CALCULATING CENTRIPETAL
ACCELERATION
A motorcycle drives around a bend with a 50meter radius at 10 m/sec. Find the motor
cycle’s centripetal acceleration and compare
it with g, the acceleration of gravity.
1.
2.
3.
4.
5.
You are asked for centripetal acceleration and a
comparison with g (9.8 m/s2).
You are given the linear speed and radius of the motion.
Use: ac = v2 ÷ r
4. Solve: ac = (10 m/s)2 ÷ (50 m) = 2 m/s2
The centripetal acceleration is about 20%, or 1/5 that 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 often incorrectly
identified as 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.
CHAPTER 8: MOTION IN CIRCLES
• 8.1 Circular Motion
• 8.2 Centripetal Force
• 8.3 Universal Gravitation and Orbital Motion
INV 8.3 UNIVERSAL GRAVITATION AND
ORBITAL
MOTION
Investigation Key Question:
How strong is gravity in other
places in the universe?
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)
CALCULATING THE WEIGHT OF A PERSON
ON THE MOON
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 Moon’s surface.
1.
2.
3.
4.
You are asked to find a person’s weight on the Moon.
You are given the radius and the masses.
Use: Fg = Gm1m2 ÷ r 2
Solve:
8.3 ORBITAL MOTION
• A satellite is an object that
is bound by gravity to
another object such as a
planet or star.
• An orbit is the path
followed by a satellite. The
orbits of many natural and
man-made satellites are
circular, or nearly circular.
8.3 ORBITAL MOTION
• The motion of a satellite is
closely related to projectile
motion.
• If an object is launched
above Earth’s surface at a
slow speed, it will follow a
parabolic path and fall
back to Earth.
 At a launch speed of about 8 kilometers per second,
the curve of a projectile’s path matches the curvature
of the planet.
8.3 SATELLITE MOTION
• The first artificial satellite, Sputnik I, which translates as
“traveling companion,” was launched by the former
Soviet Union on October 4, 1957.
• For a satellite in a circular orbit, the force of Earth’s
gravity pulling on the satellite equals the centripetal
force required to keep it in its orbit.
8.3 ORBIT EQUATION
• The relationship between a satellite’s orbital radius, r,
and its orbital velocity, v is found by combining the
equations for centripetal and gravitational force.
8.3 GEOSTATIONARY ORBITS
• To keep up with Earth’s rotation, a
geostationary satellite must travel
the entire circumference of its orbit
(2π r) in 24 hours, or 86,400 seconds.
• To stay in orbit, the satellite’s radius
and velocity must also satisfy the
orbit equation.
Use of HEO
• All geostationary satellites must
orbit directly above the equator.
• This means that the geostationary
“belt” is the prime real estate of
the satellite world.
• There have been international
disputes over the right to the
prime geostationary slots, and
there have even been cases
where satellites in adjacent slots
have interfered with each other.