introduction to circular motion - northern · pdf filea 72-kg woman rides a bicycle in a...

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Vocabulary

Term Definition Centripetal Force The Force on an object that keeps it moving in a circular

path. The force is directed towards the center of the circle.

Centripetal Acceleration

The acceleration of an object moving in a circular path is directed towards the center.

Rotate An object rotates when it moves around its center axis. (i.e. – the Earth rotates on its axis)

Revolve An object revolves around a point that is not part of the

system. (i.e. – The Earth revolves around the Sun)

Linear Speed How fast an object travels in a linear direction, by using the circumference and dividing by time.

(Relate this to a spinning car tire and how far the car travels along the ground)

Angular Speed How fast an object spins. Found by taking the number of

rotations or revolutions and dividing by time.

Center of Gravity The location where mass is believed to be located based on the shape of the object and how gravity acts on it.

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Force Relationships

1. FORCE AND MASS a. An object swung in a uniform circle with constant speed requires a certain amount of force.

b. An object with twice the mass swung in the same circle with the same speed requires twice as much force.

c. An object with three times the mass swung in the same circle with the same speed requires three times as much force.

d. From this we can conclude that the force required to keep an object in uniform circular motion is

X directly proportional to the mass of the object. ___ inversely proportional to the mass of the object. ___ some other proportionality.

m F

2m 2F

3m 3F

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2. FORCE AND RADIUS a. An object swung in a uniform circle with constant speed requires a certain amount of force.

b. The same object swung with the same speed in a circle with twice the radius requires half as much force.

c. The same object swung with the same speed in a circle with half the radius requires twice as much force. Draw your own image:

d. From this we can conclude that the force required to keep an object in uniform circular motion is ___ directly proportional to the radius of the circle. X inversely proportional to the radius of the circle. ___ some other proportionality.

3. FORCE AND SPEED a. An object swung in uniform circular motion with constant speed requires a certain amount of force. b. The same object swung with twice the speed in the same circle requires four times as much force. c. The same object swung with three times the speed in the same circle requires nine times as much force d. From this we can conclude that the force required to keep an object in uniform circular motion is

___ directly proportional to the speed of the object. ___ inversely proportional to the speed of the object. X some other proportionality.

4. Write a proportionality that incorporates all the findings regarding centripetal force.

A Force required to keep an object moving in circular motion is directly proportional to the mass of the object, and inversely proportional to the radius of the circle. The force also undergoes a squared relationship with the velocity of the swing.

m F

r

m F/2

2r

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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

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. Velocity2 3. Radius In the picture below Stewie swings Peter in a circle. Label the following with arrows:

a) the direction of the centripetal force, b) the direction of Peter’s acceleration, c) the direction Peter would travel if Stewie let go.

c

ac

v2

R

b

a

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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 perpendicular / tangent to the circular path along which the car is turning.

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 Given Relationship Solution

Centripetal Force Fc

m = 300 kg r = 20 m

v = 3 m/s 𝐹𝑐 =

𝑚𝑣2

𝑟=

300(32)20

Fc = 135 N

B. What is the centripetal acceleration? Looking For Given Relationship Solution

Centripetal Acceleration ac

r = 20 m v = 3 m/s 𝑎𝑐 =

𝑣2

𝑟=

(32)20

ac = .45 m/s2

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 Given Relationship Solution Centripetal Force

Fc m = 10 kg

r = 3 m v = 8.1 m/s

𝐹𝑐 =𝑚𝑣2

𝑟=

10(8.12)3

Fc = 218.7 N

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 Given Relationship Solution Centripetal Force

Fc m = 235 kg

r = 25 m v = 12.126 m/s

𝐹𝑐 =𝑚𝑣2

𝑟=

235(12.132)25

Fc = 1,382.17 N

ac

v2

R

Fc=mv2/R

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B. What is the centripetal acceleration? Looking For Given Relationship Solution


r = 25 m v = 12.126 m/s 𝑎𝑐 =

𝑣2

𝑟=

(12.1262)25

ac = 5.88 m/s2

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 Given Relationship Solution


m = 72 kg r = 11,549.88 m

v = 0.25 m/s 𝐹𝑐 =

𝑚𝑣2

𝑟=

72(0.252)11,549.88

Fc = 0.000389 N

B. What is the centripetal acceleration?



r = 11549.88 m v = .25 m/s 𝑎𝑐 =

𝑣2

𝑟=

(. 252)11,549.88

ac = 0.0000054 m/s2

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 Given Relationship Solution


m = 25 kg r = 2.4 m

v = 2.8 m/s 𝐹𝑐 =

𝑚𝑣2

𝑟=

25(2.82)2.4

Fc = 81.667 N

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 Given Relationship Solution


m = 65 kg r = 44 m

v = 12 m/s 𝐹𝑐 =

𝑚𝑣2

𝑟=

65(122)44

Fc = 212.72 N

B. Calculate the centripetal acceleration?



r = 44 m v = 12 m/s 𝑎𝑐 =

𝑣2

𝑟=

(122)44

ac = 3.272 m/s2

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.



m = 1100 kg r = 150 m

v = 27 m/s 𝐹𝑐 =

𝑚𝑣2

𝑟=

1100(272)150

Fc = 5,346 N

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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.



m = 0.4 kg r = 0.8 m v = 4 m/s

𝐹𝑐 =𝑚𝑣2

𝑟=

0.4(42)0.8

Fc = 8 N

B. Calculate the centripetal acceleration of the stone.



r = 0.8 m v = 4 m/s 𝑎𝑐 =

𝑣2

𝑟=

(42)0.8

ac = 20 m/s2

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 Given Relationship Solution


r = 25 m v = 10 m/s 𝑎𝑐 =

𝑣2

𝑟=

(102)25

ac = 4 m/s2

B. Determine the centripetal force acting on the car.



m = 900 kg r = 25 m

v = 10 m/s 𝐹𝑐 =

𝑚𝑣2

𝑟=

900(102)25

Fc = 3,600 N

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 .

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Rotate vs. Revolve What is the difference between rotating and revolving?

• An object rotates about its axis when the axis is internal. List three examples of an object that rotates:

• An object revolves when it moves around an external axis. List three examples of an object that revolves:

Angular Speed vs. Linear Speed

• Angular speed is the rate at which something turns. The rpm, or rotation per minute, is commonly used for angular speed.

𝐴𝑛𝑔𝑢𝑙𝑎𝑟 𝑆𝑝𝑒𝑒𝑑 = # 𝑟𝑜𝑡𝑎𝑡𝑖𝑜𝑛𝑠

𝑡𝑖𝑚𝑒

Or

𝐴𝑛𝑔𝑢𝑙𝑎𝑟 𝑆𝑝𝑒𝑒𝑑 = # 𝑟𝑒𝑣𝑜𝑙𝑢𝑡𝑖𝑜𝑛𝑠

𝑡𝑖𝑚𝑒

• Linear speed is the distance traveled per unit of time.

𝐿𝑖𝑛𝑒𝑎𝑟 𝑆𝑝𝑒𝑒𝑑 =2𝜋𝑅(# 𝑟𝑜𝑡𝑎𝑡𝑖𝑜𝑛𝑠)

𝑡𝑖𝑚𝑒

Or

𝐿𝑖𝑛𝑒𝑎𝑟 𝑆𝑝𝑒𝑒𝑑 =2𝜋𝑅(# 𝑟𝑜𝑡𝑎𝑡𝑖𝑜𝑛𝑠)

𝑡𝑖𝑚𝑒

Angular vs. Linear Speed How is the angular and linear speed of the Burt and Ernie below similar or different?

Each point on a rotating object has the same angular speed thus Burt and Ernie have the same angular speed. The linear speed of a person on a merry-go-round is the distance traveled around the circle divided by the time. The linear speed depends on the radius of the circle in which the person moves. Burt moves in a circle with the largest radius, so his linear speed is the fastest. Two people sitting at different places on the same merry go-round always have the same angular speed. But the person sitting farther from the center has the faster linear speed. Class Work

1. A wheel makes 10 revolutions in 5 seconds. Find its angular speed in rotations per second. Looking For Given Relationship Solution

Angular Speed

# revolutions = 16 t = 4 s 𝑎𝑛𝑔. 𝑠𝑝𝑒𝑒𝑑 =

(# 𝑅𝑒𝑣)𝑡

= (16)

4

Angular speed = 4 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 Given Relationship Solution

Angular Speed # revolutions = 15 t = 3 min 𝑎𝑛𝑔. 𝑠𝑝𝑒𝑒𝑑 =


= (15)

3

Angular speed = 5 rpm

(b) What is your linear speed in meters per second?


Linear Speed

# revolutions = 15 t = 180 s r = 3 m

𝑙𝑖𝑛. 𝑠𝑝𝑒𝑒𝑑 =2𝜋𝑟(#𝑅𝑒𝑣)

𝑡

= 2𝜋(3)(15)

180

Linear speed = 1.57 m/s

3. A compact disc completes 60 rotations in 5 seconds.

a. What is its angular speed? Looking For Given Relationship Solution

Angular Speed # revolutions = 60 t = 5 s 𝑎𝑛𝑔. 𝑠𝑝𝑒𝑒𝑑 =


= (60)

5


Burt

Ernie

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.


Linear Speed

# revolutions = 4 t = 1 s

r = .06 m


𝑡

= 2𝜋(.06)(4)

1


5. A merry-go-round makes 18 rotations in 3 minutes. What is its angular speed in rpm?




= (18)

3


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 Given Relationship Solution

Linear Speed

# revolutions = 1 t = 10 s r = 2 m


𝑡

= 2𝜋(2)(1)

10


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 Given Relationship Solution



= (12)

3


8. Mao watches a merry-go-round as it turns 27 times in 3 minutes. The angular speed of the merry-go-round is

____ rpm. Looking For Given Relationship Solution



= (27)

3


HomeWork 1. A wheel makes 20 revolutions in 5 seconds. Find its angular speed in rotations per second.


Angular Speed # revolutions = 20 t = 5 seconds 𝑎𝑛𝑔. 𝑠𝑝𝑒𝑒𝑑 =


= (20)

5


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?




= (15)

3


(b) What is your linear speed in meters per second? Looking For Given Relationship Solution

Linear Speed

# revolutions = 15 t = 3 min = 180 s

r = 2.5 m


𝑡

= 2𝜋(2.5)(15)

180


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 Given Relationship Solution

Linear Speed

# revolutions = 1 t = 1 s

r = 0.06 m


𝑡

= 2𝜋(.06)(1)

1


4. A merry-go-round makes 30 rotations in 3 minutes. What is its angular speed in rpm?




= (30)

3


Introduction to Circular Motion Research Question What factors will affect the circular motion of an object? Hypothesis Predict what factors you think will affect the motion of an object in circular motion? Materials 8 elastic bands, balance, meterstick, plastic bottle marked at the 150 mL level, 14 oz. plastic drinking cup with three equally spaced holes below the rim, stopwatch. Procedure

1. The device shown is referred to as a cupsling.

2. Carefully measure 50 mL of water into the plastic cup. Make sure that no water spills.

3. Place the cupsling on a balance, and record its mass with appropriate units.

4. Make sure that the area is clear of obstacles and slowly spin the full cupsling about you in a full circle. Slightly increase the speed until you can spin it so that the cup moves in a horizontal circle. Try to see how slowly you can spin the cupsling and still consistently maintain a horizontal circle. Be careful not to spill or splash any water.

5. With the stopwatch, time 10 complete circles of the cup as you swing it slowly around in a horizontal

circle. Record this measurement in the data table.

6. Estimate the radius of the cup’s horizontal path at this speed using the meterstick. Get as precise an estimate as possible. Record this measurement in the data table.

7. Repeat the above steps, spinning the cupsling faster than before but not so fast that the bands will break.

Remember to consistently maintain a horizontal circle throughout the experiment. Record this measurement in the data table.

8. Empty the cup. Refill it with 100 mL of water. Find the mass of the cupsling with the water and record

this mass in the data table.

9. Repeat the procedure, measuring the time for 10 circles and the radius when you swing it slowly around in a horizontal circle. Record the measurements in the data table.

10. Repeat the steps above for 150 mL, 200 mL, and 250 mL of water.

Data Table: Trial Mass of cupsling

and water (kg) Time for 10 cycles

(s) Period (s) Radius (m)

1 2 3 4 5

Analysis Questions

1. Did you need to exert a force on the elastic band to start spinning the cupsling from rest? 2. Did you need to continue exerting a force on the elastic band to keep it spinning at a constant speed?

How did you know?

3. When the cupsling moved in a circle, it was changing direction all the time. What caused the cupsling to change direction?

4. When the cupsling moved in a circle at constant speed, did it accelerate? Explain.

5. Where do you think the cup would go if the band were released while the cup was spinning? Draw an

arrow on the top view of your circular path to represent the motion of the cupsling if release at the position shown:

6. What happened to the length of the elastic band as you increased the force to spin the cupsling in a horizontal circle?

7. What happened to the length of the elastic band as the speed increased?

8. What happened to the force of the elastic band as the speed increased?

9. What happened to the length of the elastic band when you increased the mass in the cup?

10. How did the increase in mass affect the force on the elastic band?

11. If a mass moves in a straight line and more mass is added, does the inertia increase, decrease, or stay the

same?

12. Do you think that the same thing happens to a body in circular motion? Explain.

Activity – The Flying Pig

Research Question: What factors affect the angular and linear speed of a flying pig? Procedure:

1. Setup the Flying Pig. Be careful not to damage their delicate wings as you click them into their fixed-wing position.

2. Carefully hold the pig by its body and give it a slight shove about 30 degrees from the vertical, just enough so that the pig “flies” in a circle. The goal is to launch the pig tangent to the circle of flight. It is better to launch it too easy than too hard. If the pig does not fly in a stable circle in 10 seconds or so, carefully grab it and try launching it again.

3. Once the pig is up and flying in a circle of constant radius, measure the radius of the circle as accurately as you can.

4. Count the number of revolutions the pig makes for the time intervals in the data table below. 5. Calculate the following for each trial:

a. Angular speed: The number of revolutions per time. If the pig makes 2 revolutions in one second then the angular speed is 2 revolutions per second. If the pig makes 8 revolutions in 4 seconds then the angular speed is 8 revolutions/4 seconds = 2 revolutions per second.

b. Linear Speed (v): 2πR/t c. Centripetal Acceleration (ac): v2/R

Data Table

Pig # Revolutions Time (s)

Angular Speed

(revolutions per second)

Linear Speed (m/s)

Centripetal Acceleration

(m/s2)

1 30 2 30 3 30 4 30 5 30 6 30 7 30

Center of Gravity

For a given body, the center of mass is the average location 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 CG for short, when consideration of this position is necessary. The CG 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 hanging. The CG may be a point where no mass actually 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.

http://schools.wikia.com/wiki/Image:Mid-Air.jpg

Directions: Answer the following questions in complete sentences. 1. Why does the Leaning Tower of Pisa not topple? Use the picture below for a clue.

The tower does not topple because the Center of Gravity is centered over the base of the object. If the mass were tilted more, and the CG was past the base, then it would topple over.

\ 2. How can you design objects to reduce the likelihood of tipping?

You can reduce the likelihood of tipping by designing objects that are geometrically similar in shape and have the CG at the center of the object.

3. Decide which of the following trucks will tip over. Explain why.

The truck labeled A will fall over because when drawing a line straight down for Center of Gravity, the line is outside of the wheel supports. The trucks labeled B and C will not tip over because when drawing a line straight down for Center of Gravity, the line is within the wheel supports.

4. Will the pipe below fall over? Explain why or why not.

The pipe will not fall over because if you draw a line straight down through the Center of Gravity, it is found within the area of contact for the pipe.

Lab - Center of Gravity Physics

Research Question What factors determine the stability of an object (ie whether or not an object will topple)? Write a prediction in response to the research question. Be sure to supply your reasoning, which should include a definition of the center of gravity and any examples that you can provide to support your ideas. Some examples could be why the Leaning Tower of Pisa doesn’t rotate and topple over. Another could be why you can balance a toy bird on your finger.

Procedure 1. You will be moving around with a partner to the 10 different stations listed on the back of this page. At

each station, follow the instructions printed on the paper. 2. In each case, you will first generate a prediction or hypothesis. Make sure to briefly explain your

reasoning!

3. Record your observations in a systematic and organized way. You should make annotated drawings, an observations table with a checklist, or any other method that will work for you.

4. Go to as many stations as possible during the lab period. Some take longer than others so please do not

rush a group that is working slowly. When possible, there are double set-ups, so wait your turn to get to an empty station.

introduction to circular motion - northern · pdf filea 72-kg woman rides a bicycle in a...

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