Download Introduction to Circular Motion

Vocabulary
Term
Centripetal Force
Definition
A force that causes an object to move in a circle.
Centrifugal Force
The effect of inertia on an object moving in a circle.
Centripetal
Acceleration
The rate of change of speed of an object moving in a circle.
Rotate
To spin around an axis of rotation that passes through an
object.
Revolve
To move around, or orbit, an external axis.
Linear Speed
The distance traveled per unit of time.
Angular Speed
The rate at which an object rotates or revolves.
Center of Gravity
The average position of an object’s weight.
Centripetal Force (Fc)
What is centripetal force?
•Some physical force pushing or pulling the object towards the center of the circle.
•The word "centripetal" is merely an adjective used to describe the direction of the force.
• Without the centripetal force, the object will move in a straight line.
• Centripetal force is any force that causes an object to move in a circle.
• To calculate centripetal force:
Fc=mv2/R
• To calculate centripetal acceleration:
ac=v2/R
v2
ac
R
Give three examples of centripetal force.
 As a car makes a turn, the force of friction acting upon the turned wheels of the car provide the
centripetal force required for circular motion.
 As the moon orbits the Earth, the force of gravity acting upon the moon provides the centripetal
force required for circular motion.
 What three factors affect the centripetal force of an object moving in a circle?
1. mass
2. Velocity
3. Radius
 In the picture below Stewy swings Peter in a circle. Label the following with arrows: the direction
of the centripetal force, the direction of Peter’s acceleration, and the direction Peter would travel if
Stewy let go.
v
Fc
ac
But what about centrifugal forces?
• There is no such thing! The sensation of an outward force and an outward acceleration is a false sensation.
• For example, if you are in a car make a right turn, while the car is accelerating inward, your body continues
•
in a straight line. If you are sitting on the passenger side of the car, then eventually the outside door of the
car will hit you as the car turns inward. In reality, you are continuing in your straight-line inertial path
tangent to the circle while the car is accelerating out from under you.
It is the inertia of your body - the tendency to resist acceleration - which causes it to continue in its forward
motion. There is no physical object capable of pushing you outwards. You are merely experiencing the
tendency of your body to continue in its path tangent to the circular path along which the car is turning.
2
Fc=mv /R
v2
ac
R
Class Work
1. A 300-kg waterwheel rotates about its 20-m radius axis at a rate of 3 meters per second.
A. What is the centripetal force requirement?
Looking For
Centripetal force
Given
M=300 kg
R=20 m
V= 3 m/s
Relationship
Given
v=3 m/s
R=20 m
Relationship
Solution
135 N
mv2/R
B. What is the centripetal acceleration?
Looking For
Centripetal acceleration
Solution
0.45 m/s2
v2/R
2. A 10-kg mass is attached to a string and swung horizontally in a circle of radius 3-m. When the speed of the mass
reaches 8.1 m/s, what is the centripetal force requirement?
Looking For
Centripetal force
Given
M=10 kg
R=3 m
V= 8.1 m/s
Relationship
Solution
218.7 N
mv2/R
3. A motorcycle travels 12.126 m/s in a circle with a radius of 25.0 m.
A. How great is the centripetal force that the 235-kg motorcycle experiences on the circular path?
Looking For
Centripetal force
Given
M=235 kg
R=25 m
V= 12.126 m/s
Relationship
Given
Relationship
Solution
1382.17 N
mv2/R
B. What is the centripetal acceleration?
Looking For
Solution
Centripetal acceleration
v=12.126 m/s
R=25 m
5.88 m/s2
v2/R
Group Work
4. A 72-kg woman rides a bicycle in a 75.57-km circumference circle at a rate of 0.25 m/s.
A. What is the centripetal force experienced by the woman?
Looking For
Centripetal force
Given
M=72 kg
R=75570 m
V= .25 m/s
Relationship
Solution
.000059 N
mv2/R
B. What is the centripetal acceleration?
Looking For
Centripetal acceleration
Given
v=.25 m/s
R=75570 m
Relationship
Solution
.000000827 m/s2
v2/R
5. A 25-kg mass swings on a string with a length of 2.4-m so that the speed at the bottom point is 2.8 m/s. Calculate
the centripetal force.
Looking For
Centripetal force
Given
M=25 kg
R=2.4 m
V= 2.8 m/s
Relationship
Solution
81.7 N
mv2/R
6. A 65-kg mass swings on a 44-m long rope. If the speed at the bottom point of the swing is 12 m/s,
A. What is the centripetal force experienced by the mass?
Looking For
Centripetal force
Given
M=65 kg
R=44 m
V= 12 m/s
Relationship
Solution
212.7 N
mv2/R
B. Calculate the centripetal acceleration?
Looking For
Centripetal acceleration
Given
v=12 m/s
R=44 m
Relationship
Solution
3.27 m/s2
v2/R
7. Determine the centripetal force acting on an 1100-kg car that travels around a highway curve of radius 150 m at
27 m/s.
Looking For
Centripetal force
Given
M=1100 kg
R=150 m
V= 27 m/s
Relationship
Solution
5346 N
mv2/R
8. Roxanne is making a strawberry milkshake in her blender. A tiny, 0.0050 kg strawberry is rapidly spun around
the inside container with a speed of 14.0 m/s, held by a centripetal force of 10.0 N. What is the radius of the
blender at this location.
Looking For
radius
Given
M=.005 kg
F=10 N
Relationship
mv2/R
Solution
.098 m
V= 14 m/s
HomeWork
1. The diagram below represents a 0.40-kilogram stone attached to a string. The stone is moving at a
constant speed of 4.0 meters per second in a horizontal circle having a radius of 0.80 meter.
A. Calculate the centripetal force acting on the stone.
Looking For
Centripetal force
Given
M=.4 kg
R=.8 m
V= 4 m/s
Relationship
Solution
8N
mv2/R
B. Calculate the centripetal acceleration of the stone.
Looking For
Centripetal acceleration
Given
v=4 m/s
R=.8 m
Relationship
Solution
20 m/s2
v2/R
2. A 900-kg car moving at 10 m/s takes a turn around a circle with a radius of 25.0 m.
A. Determine the centripetal acceleration of the car.
Looking For
Centripetal acceleration
Given
v=10 m/s
R=25 m
Relationship
Solution
4 m/s2
v2/R
B. Determine the centripetal force acting on the car.
Looking For
Centripetal force
Given
M=900 kg
R=25 m
V= 10 m/s
Relationship
Solution
3600 N
mv2/R
3. According to the diagram of the plane below, the direction of the centripetal force on the airplane is directed
toward: D
4. According to the diagram of the plane below, the direction of the acceleration on the airplane is directed toward:
D.
5. According to the diagram of the plane below, the direction the plane would travel if a centripetal force was no
longer applied is toward: A
Class Work
1. A wheel makes 10 revolutions in 5 seconds. Find its angular speed in rotations per second.
Looking For
Angular speed
Given
Revs = 10
Time = 5 s
Relationship
Revolutions/time
Solution
2 rps
2. You are sitting on a merry-go-round at a distance of 3 meters from its center. It spins 15 times in 3
minutes. (a) What is your angular speed in revolutions per minute?
Looking For
Angular speed
Given
Revs = 15
Time = 3 min
Relationship
Revolutions/time
Solution
5 rpm
(b) What is your linear speed in meters per second?
Looking For
Linear speed
Given
R=3 m
# of revs = 15
Time = 180 s
Relationship
Solution
2πR(# of revolutions) / time
1.57 m/s
Relationship
Revolutions/time
Solution
12 rps
3. A compact disc completes 60 rotations in 5 seconds.
a. What is its angular speed?
Looking For
Angular speed
Given
Revs = 60
Time = 5 s
Group Work
4. A compact disc has a radius of 0.06 meters. If the cd rotates 4 times per second, what is the linear speed
of a point on the outer edge of the cd? Give your answer in meters per second.
Looking For
Linear speed
Given
R=.06 m
# of revs = 4
Time = 1 s
Relationship
Solution
2πR(# of revolutions) / time
1.5 m/s
5. A merry-go-round makes 18 rotations in 3 minutes. What is its angular speed in rpm?
Looking For
Angular speed
Given
Revs = 18
Time = 3 min
Relationship
Revolutions/time
Solution
6 rpm
6. Dwayne sits two meters from the center of a merry-go-round. If the merry-go-round makes one
revolution in 10 seconds, what is Dwayne’s linear speed?
Looking For
Linear speed
Given
R=2 m
# of revs = 1
Time = 10 s
Relationship
Solution
2πR(# of revolutions) / time
1.256 m/s
7. Find the angular speed of a ferris wheel that makes 12 rotations during 3 minute ride. Express your
answer in rotations per minute.
Looking For
Angular speed
Given
Revs = 12
Time = 3 min
Relationship
Revolutions/time
Solution
4 rpm
8. Mao watches a merry-go-round as it turns 27 times in 3 minutes. The angular speed of the merry-go-round is 9
rpm.
9. Calculate the angular speed of a bicycle wheel that makes 240 rotations in 6 minutes.
Looking For
Angular speed
Given
Revs = 240
Time = 6 min
Relationship
Revolutions/time
Solution
40 rpm
HomeWork
1. A wheel makes 20 revolutions in 5 seconds. Find its angular speed in rotations per second.
Looking For
Angular speed
Given
Revs = 20
Time = 5 s
Relationship
Revolutions/time
Solution
4 rps
2. You are sitting on a merry-go-round at a distance of 2.5 meters from its center. It spins 15 times in 3
minutes. (a) What is your angular speed in revolutions per minute?
Looking For
Angular speed
Given
Revs = 15
Time = 3 s
Relationship
Revolutions/time
Solution
5 rpm
Relationship
Solution
2πR(# of revolutions) / time
1.3 m/s
(b) What is your linear speed in meters per second?
Looking For
Linear speed
Given
R=2.5 m
# of revs = 15
Time = 180 s
3. A compact disc has a radius of 0.06 meters. If the cd rotates once every second, what is the linear speed
of a point on the outer edge of the cd? Give your answer in meters per second.
Looking For
Linear speed
Given
R=.06 m
# of revs = 1
Time = 1 s
Relationship
Solution
2πR(# of revolutions) / time
.3768 m/s
4. A merry-go-round makes 30 rotations in 3 minutes. What is its angular speed in rpm?
Looking For
Angular speed
Given
Revs = 30
Time = 3 min
Relationship
Revolutions/time
Solution
10 rpm
Center of Gravity
For a given body, the center of mass is the average position of all the mass that makes up the object. A
symmetrical object like a ball can be thought of as having all of its mass concentrated at its geometric
center; by contrast, an irregularly shaped object such as a baseball bat has more of its mass toward one end.
Center of gravity is the same thing as the center of mass, except specifically referring to an object under the
influence of gravity. The terms are effectively synonymous and we will use the abbreviation COG or CG for
short, when consideration of this position is necessary.
The COG of a uniform object, such as a meter stick, is at its center, for the stick acts as though its entire
weight were concentrated there. Support at that single point supports the whole stick. Holding an object
provides a simple method of locating its CG.
The CG of any freely suspended object lies directly beneath (or at) the point of support.
The CG may be a point where no point exists. For example, the center of mass of a ring or a hollow sphere
is at the geometrical center where no matter exists. Similarly, the center of mass of a boomerang is outside
the physical structure, not within the material making up the boomerang.
Read pages 151-154 then answer the following questions
Who discovered Gravity? Newton
What keeps the moon in orbit around the Earth? Gravity
What three factors affect the weight of an object?
1. Mass
2. Mass
3. distance
An increase in which of the three factors above will increase your weight the most?
mass
Why do you not feel the gravitational force between you and your textbook?
The masses are so small compared to that of the planets.
If you double the mass of an object what happens to the gravitational force?
It doubles.
The distance between objects, measured from center to center, is also important when calculating gravitational
force. The closer objects are to each other, the greater the force between them. The farther apart, the smaller the
force. The decrease in gravitational force is related to the square of the distance.
Doubling the distance divides the force by four. If you are twice as far from an object, you feel one-fourth the
gravitational force. Tripling the distance divides the force by nine. If you are three times as far away, the force
is one-ninth as strong.
What is the Law of Universal Gravitation?
Gives the relationship between gravitational force, mass, and distance.