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.
Speed and Velocity
Review:
1. A quantity that is fully described by magnitude alone is a ___________ quantity. A quantity that is fully described
by both magnitude and direction, is a ___________ quantity.
a. scalar, vector b. vector, scalar
2. Speed is a ____________ quantity. Velocity is a ____________ quantity.
a. scalar, vector b. vector, scalar
c. scalar, scalar
d. vector, vector
3. State the equation for calculating the average speed of an object: distance/time
Circular Motion:
4. An object that moves uniformly in a circle can have a constant ___________________ but a changing
___________________.
a. speed, velocity
b. velocity, speed
5. The direction of a velocity vector is always ______. Circle all that apply.
a. in the same direction as the net force that acts upon it
b. in the opposite direction as the net force that acts upon it
c. in the same direction as the object is moving
d. in the opposite direction as the object is moving
e. ... none of these!
6. True or False:
The direction of the velocity vector of an object at a given instant in time depends on whether the object is
speeding up or slowing down.
7. For an object moving in uniform circular motion, the velocity vector is directed _____.
a. radially inwards towards the center of the circle
b. radially outwards away from the center of the circle
c. in the direction of the tangent line drawn to the circle at the object's location
8. Use your average speed equation to determine the speed of ... . (Given: Circumference = 2•PI•R)
a. ... a rider on a carousel ride that makes a complete revolution around the circle (radius = 10.6meter) in 17.3 seconds. PSYW
Looking For
speed
Given
Distance=2πR=2π(10.6)
Time=17.3 seconds
Relationship
Speed = distance/time
Solution
3.85 m/s
b. ... your clothes that are plastered to the wall of the washing machine during the spin cycle. The
clothes make a complete revolution around a 0.35 meter circle in 0.285 seconds. PSYW
Looking For
Given
Relationship
Solution
speed
Distance=2πR=2.19 meter
Time=0.285 seconds
Speed = distance/time
7.72 m/s
9. A roller coaster car is traveling over the crest of a hill and is at the location shown. A side view is shown at the
right. Draw an arrow on the diagram to indicate the direction of the velocity vector.
10.
Circular Motion and Inertia
Review Questions:
1. Newton's first law states: An object at rest will remain at rest.
An object in motion will stay in motion at constant speed in a straight line
unless acted upon by an unbalanced force.
2. Inertia is ... the tendency of an object to resist changes to its state of motion.
Applications of Newton's First Law to Motion in Circles:
The diagram below depicts a car making a right hand turn. The driver of the car is represented by the
circled X. The passenger is represented by the solid circle. The seats of the car are vinyl seats and have
been greased down so as to be smooth as silk. As would be expected from Newton's law of inertia, the
driver continues in a straight line from the start of the turn until point A. The path of the driver is shown:
B
F
Direction of
force
Rex Things and Doris Locked are out on a date. Rex makes a rapid right-hand turn. Doris begins sliding across
the vinyl seat (which Rex had waxed and polished beforehand) and collides with Rex. To break the awkwardness
of the situation, Rex and Doris begin discussing the physics of the motion that was just experienced. Rex suggests
that objects that move in a circle experience an outward force. Thus, as the turn was made, Doris experienced an
outward force that pushed her towards Rex. Doris disagrees, arguing that objects that move in a circle experience
an inward force.
In this case, according to Doris, Rex traveled in a circle due to the force of his door pushing him inward. Doris did
not travel in a circle since there was no force pushing her inward; she merely continued in a straight line until she
collided with Rex. Who is correct? ________ Argue one of these two positions.
Doris would be correct. There is no such thing as an “outward” force. The reason why this happens is due to an
object’s inertia or the tendency to continue moving in a straight line.
Noah Formula guides a golf ball around the outside rim of the green at the Hole-In-One PuttPutt Golf Course. When the ball leaves the rim, which path (1, 2, or 3) will the golf ball
follow? Explain why.
The ball will follow path 2 due to the ball’s inertia (the tendency to continue moving in a
straight line).
Suppose that you are a driver or passenger in a car and you travel over the top of a small hill in the road at a high
speed. As you reach the crest of the hill, you feel your body still moving upward; your gluts might even be
pulled off the car seat. It might even feel like there is an upward push on your body. This upward sensation
is best explained by the ______.
a. tendency of your body to follow its original upward path
b. presence of an upward force on your body
c. presence of a centripetal force on your body
d. presence of a centrifugal force on your body
Darron Moore is on a barrel ride at an amusement park. He enters the barrel and stands on a
platform next to
the wall. The ride operator flips a switch and the barrel begins spinning
at a high rate. Then the operator
flips another switch and the
platform drops out from
under the feet of the riders. Darron is plastered
to the wall of the barrel. This sticking to the
wall phenomenon is explained by the fact that ________.
a. the ride exerts an outward force on Darron which pushes him outward against the wall
b. Darron has a natural tendency to move tangent to the circle but the wall pushes him inward
c. air pressure is reduced by the barrel's motion that causes a suction action toward the wall
d. the ride operator coats the wall with cotton candy that causes riders to stick to it
Always take time to reflect upon your own belief system that governs how you interpret the physical world. Be aware of
your personal "mental model" which you use to explain why things happen. The idea of this physics course is not to
acquire information through memorization but rather to analyze your own preconceived notions about the world and to
dispel them for more intelligible beliefs. In this unit, you will be investigating a commonly held misconception about the
world - that motion in a circle is caused by an outward (centrifugal) force. This misconception or wrong belief is not
likely to be dispelled unless you devote some time to reflect on whether you believe it and whether it is intelligible. After
considering more reasonable beliefs, you will be more likely to dispel the belief in a centrifugal force in favor of a belief
in an inward or centripetal force.
Speed and Velocity
1. What is uniform circular motion?
The motion of an object in a circle with a constant or uniform speed.
2. What is the formula to calculate the average speed for an object traveling in a circular path?
3. How are average speed and radius related?
Average speed and radius are directly related. If you double the radius then the
If you triple the radius, then the average
speed triples, and so on.
4. How do speed and velocity differ?
Speed is a scalar quantity and velocity is a vector quantity. Thus, speed has
includes both magnitude and direction.
average speed doubles.
magnitude only while velocity
5. Draw a picture showing the direction of an object’s velocity when traveling in a circular path.
6. Used words to describe the direction of the velocity vector.
The directions of the velocity vector at every instant is in a direction tangent to the
circle.
7. Summarize the differences between an object’s speed and velocity while moving in uniform circular motion.
An object moving in uniform circular motion is moving around the perimeter of the circle with a
constant speed. While the speed of the object is constant, its velocity is changing. Velocity, being a vector, has a
constant magnitude but a changing direction. The direction is always directed tangent to the circle and as the
object turns the circle, the tangent line is always pointing in a new direction.
Acceleration
1. What is a common misconception about the speed of an object moving in a circle?
An object moving in uniform circular motion is moving in a circle with a uniform or constant speed. The
velocity vector is constant in magnitude but changing in
direction. Because the speed is constant for such a
motion, many students have the
misconception that there is no acceleration. "After all," they might say, "if I
were driving a car in a circle at a constant speed of 20 mi/hr, then the speed is neither
decreasing nor
increasing; therefore there must not be an acceleration." At the center
of this common student
misconception is the wrong belief that acceleration has to
do with speed and not with velocity.
2. How does an object accelerate when it moves in a circle if its speed is constant?
An accelerating object is an object that is changing its velocity. And since velocity is a
vector that has
both magnitude and direction, a change in either the magnitude or the direction constitutes a change in the
velocity. For this reason, it can be safely
concluded that an object moving in a circle at constant speed is
indeed accelerating. It is accelerating because the direction of the velocity vector is changing.
3. Draw a picture showing the direction of an object’s acceleration when traveling in a circular path.
4. What type of device is used to measure the acceleration of an object?
An accelerometer is used to measure the acceleration of an object.
5. Identify three controls on an automobile that allow the car to be accelerated.
The three controls that allow the car to be accelerated are the brakes, gas pedal and steering wheel.
The Centripetal Force Requirement
1.
What does the word “centripetal” mean?
Centripetal means center seeking.
2. Explain how inertia relates to the motion of an object traveling in a circle.
According to Newton's first law of motion, it is the natural tendency of all moving objects to continue in
motion in the same direction that they are moving ... unless some form of unbalanced force acts upon the object
to deviate its motion from its straight-line path. Moving objects will tend to naturally travel in straight lines; an
unbalanced force is only required to cause it to turn. Thus, the presence of an
unbalanced force is
required for objects to move in circles.
3. Draw a picture showing the direction of the centripetal force acting up an object when traveling in a circular
path.
4. List three real world examples of centripetal force.
 Friction
 Tension
 Gravity
The Forbidden F-Word
1. How does the word “centrifugal” differ from “centripetal?”
Centrifugal, not to be confused with centripetal, means away from the center or
outward.
2. What is a common misconception about students moving in circular motion?
The common misconception, believed by many physics students, is the notion that objects in circular
motion are experiencing an outward force.
3. What “law” explains the feeling of an outward force? Explain.
An object moving n circular motion is at all times moving tangent to the circle; the velocity vector for the
object is directedtangentially. To make the circular motion, there must be a net or unbalanced force directed
towards the center of the circle in order to deviate the object from its otherwise tangential path. This path is
an inward force - a centripetal force. That is spelled c-e-n-t-r-i-p-e-t-a-l, with a "p." The other word centrifugal, with an "f" - will be considered our forbidden F-word.
Mathematics of Circular Motion
1. What is the formula to calculate the average speed of an object moving in a circle?
2. What is the primary formula to determine the acceleration of an object moving in a circle? (Hint: the
formula involving velocity and radius.)
3. What is the primary formula used to calculate the net force acting upon an object traveling in a circle? (Hint:
the formula involves mass, velocity and radius.)
Kepler’s Three Laws
1.
Who proposed the three laws of planetary motion?
Johannes Kepler
2. What is Kepler’s 1st law of planetary motion?
The path of the planets about the sun is elliptical in shape, with the center of the sun being located at one
focus. (The Law of Ellipses)
3. What is Kepler’s 2nd law of planetary motion? Draw a picture to show how any planet sweeps out equal
areas in equal amounts of time.
An imaginary line drawn from the center of the sun to the center of the planet will sweep out equal areas
in equal intervals of time. (The Law of Equal Areas)
4. What is Kepler’s 3rd law of planetary motion?
The ratio of the squares of the periods of any two planets is equal to the ratio of the cubes of their average
distances from the sun. (The Law of Harmonies)
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.
Fc=mv2/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
mv2/R
Solution
1382.17 N
B. What is the centripetal acceleration?
Looking For
Centripetal acceleration
Given
v=12.126 m/s
R=25 m
Relationship
Solution
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
V= 14 m/s
Relationship
Solution
.098 m
mv2/R
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
Relationship
Solution
2πR(# of revolutions) / time
1.57 m/s
Relationship
Revolutions/time
Solution
12 rps
(b) What is your linear speed in meters per second?
Looking For
Linear speed
Given
R=3 m
# of revs = 15
Time = 180 s
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