Circular Motion and GravitationSection 1 Preview Section 1 Circular MotionCircular Motion Section 2 Newton’s Law of Universal GravitationNewton’s Law of.

Circular Motion and Gravitation Section 1

Preview

Section 1 Circular Motion

Section 2 Newton’s Law of Universal Gravitation

Section 3 Motion in Space

Section 4 Torque and Simple Machines


What do you think?

• Consider the following objects moving in circles:• A car traveling around a circular ramp on the highway• A ball tied to a string being swung in a circle • The moon as it travels around Earth• A child riding rapidly on a playground merry-go-round

• For each example above, answer the following:• Is the circular motion caused by a force?• If so, in what direction is that force acting? • What is the source of the force acting on each object?

Section 1Circular Motion and Gravitation

Tangential Speed (vt)

• Speed in a direction tangent to the circle

• Uniform circular motion: vt has a constant value– Only the direction changes– Example shown to the right

• How would the tangential speed of a horse near the center of a carousel compare to one near the edge? Why?


Centripetal Acceleration (ac)

• Acceleration is a change in velocity (size or direction).

• Direction of velocity changes continuously for uniform circular motion.

• What direction is the acceleration?– the same direction as v– toward the center of the circle

• Centripetal means “center seeking”


Centripetal Acceleration (magnitude)

• How do you think the magnitude of the acceleration depends on the speed?

• How do you think the magnitude of the acceleration depends on the radius of the circle?


Tangential Acceleration

• Occurs if the speed increases• Directed tangent to the circle• Example: a car traveling in a circle

– Centripetal acceleration maintains the circular motion.• directed toward center of circle

– Tangential acceleration produces an increase or decrease in the speed of the car.

• directed tangent to the circle


Click below to watch the Visual Concept.

Visual Concept

Centripetal Acceleration

Section 1Circular Motion and Gravitation

Centripetal Force (Fc)

c cF ma2

and tc

va

r

2

so tc

mvF

r


Centripetal Force

• Maintains motion in a circle• Can be produced in different

ways, such as– Gravity– A string– Friction

• Which way will an object move if the centripetal force is removed?– In a straight line, as shown on

the right


Describing a Rotating System

• Imagine yourself as a passenger in a car turning quickly to the left, and assume you are free to move without the constraint of a seat belt.– How does it “feel” to you during the turn? – How would you describe the forces acting on you during this

turn?

• There is not a force “away from the center” or “throwing you toward the door.”– Sometimes called “centrifugal force”

• Instead, your inertia causes you to continue in a straight line until the door, which is turning left, hits you.


Classroom Practice Problems

• A 35.0 kg child travels in a circular path with a radius of 2.50 m as she spins around on a playground merry-go-round. She makes one complete revolution every 2.25 s.– What is her speed or tangential velocity? (Hint: Find

the circumference to get the distance traveled.)– What is her centripetal acceleration?– What centripetal force is required?

• Answers: 6.98 m/s, 19.5 m/s2, 682 N


Now what do you think?

• Consider the following objects moving in circles:• A car traveling around a circular ramp on the highway• A ball tied to a string being swung in a circle • The moon as it travels around Earth• A child riding rapidly on a playground merry-go-round

• For each example above, answer the following:• Is the circular motion caused by a force?• If so, in what direction is that force acting? • What is the source of the force acting on each object?


What do you think?

Imagine an object hanging from a spring scale. The scale measures the force acting on the object. • What is the source of this force? What is pulling or

pushing the object downward?• Could this force be diminished? If so, how?• Would the force change in any way if the object was

placed in a vacuum?• Would the force change in any way if Earth stopped

rotating?


Newton’s Thought Experiment• What happens if you fire a

cannonball horizontally at greater and greater speeds?

• Conclusion: If the speed is just right, the cannonball will go into orbit like the moon, because it falls at the same rate as Earth’s surface curves.

• Therefore, Earth’s gravitational pull extends to the moon.


Law of Universal Gravitation

• Fg is proportional to the product of the masses (m1m2).

• Fg is inversely proportional to the distance squared (r2).– Distance is measured center to center.

• G converts units on the right (kg2/m2) into force units (N).– G = 6.673 x 10-11 N•m2/kg2


Law of Universal Gravitation


Gravitational Force

• If gravity is universal and exists between all masses, why isn’t this force easily observed in everyday life? For example, why don’t we feel a force pulling us toward large buildings?– The value for G is so small that, unless at least one of

the masses is very large, the force of gravity is negligible.


Ocean Tides

• What causes the tides?• How often do they occur? • Why do they occur at certain times? • Are they at the same time each day?


Ocean Tides

• Newton’s law of universal gravitation is used to explain the tides. – Since the water directly below the moon is closer than

Earth as a whole, it accelerates more rapidly toward the moon than Earth, and the water rises.

– Similarly, Earth accelerates more rapidly toward the moon than the water on the far side. Earth moves away from the water, leaving a bulge there as well.

– As Earth rotates, each location on Earth passes through the two bulges each day.

– Link to web


Gravity is a Field Force

• Earth, or any other mass, creates a force field.

• Forces are caused by an interaction between the field and the mass of the object in the field.

• The gravitational field (g) points in the direction of the force, as shown.


Calculating the value of g

• Since g is the force acting on a 1 kg object, it has a value of 9.81 N/m (on Earth).– The same value as ag (9.81 m/s2)

• The value for g (on Earth) can be calculated as shown below.

2 2

g E EF Gmm Gm

gm mr r


Classroom Practice Problems

• Find the gravitational force that Earth

(mE = 5.97 1024 kg) exerts on the moon

(mm= 7.35 1022 kg) when the distance between them is 3.84 x 108 m.– Answer: 1.99 x 1020 N

• Find the strength of the gravitational field at a point 3.84 x 108 m from the center of Earth.– Answer: 0.00270 N/m or 0.00270 m/s2



Imagine an object hanging from a spring scale. The scale measures the force acting on the object. – What is the source of this force? What is pulling or

pushing the object downward?– Could this force be diminished? If so, how?– Would the force change in any way if the object was

placed in a vacuum?– Would the force change in any way if Earth stopped

rotating?


What do you think?

• Make a sketch showing the path of Earth as it orbits the sun.

• Describe the motion of Earth as it follows this path.

• Describe the similarities and differences between the path and motion of Earth and that of other planets.


What do you think?

• What does the term weightless mean to you?• Have you ever observed someone in a

weightless environment? If so, when?• How did their weightless environment differ from a

normal environment?


Weight and Weightlessness

• Bathroom scale– A scale measures the downward force exerted on it.– Readings change if someone pushes down or lifts up

on you.• Your scale reads the normal force acting on you.


Apparent Weightlessness

• Elevator at rest: the scale reads the weight (600 N).• Elevator accelerates downward: the scale reads less.• Elevator in free fall: the scale reads zero because it no

longer needs to support the weight.


Apparent Weightlessness

• You are falling at the same rate as your surroundings.– No support force from the floor is needed.

• Astronauts are in orbit, so they fall at the same rate as their capsule.

• True weightlessness only occurs at great distances from any masses.– Even then, there is a weak gravitational force.



• Make a sketch showing the path of Earth as it orbits the sun.

• Describe the motion of Earth as it follows this path.

• Describe the similarities and differences between the path and motion of Earth and that of other planets.



• What does the term weightless mean to you?• Have you ever observed someone in a

weightless environment? If so, when?• How did their weightless environment differ from a

normal environment?


Simple Machines

• Change the size or direction of the input force• Mechanical advantage (MA) compares the input

force to the output force.– When Fout > Fin then MA > 1

• MA can also be determined from the distances the input and output forces move.

MA FoutFin

dindout


Click below to watch the Visual Concept.

Visual Concept

Overview of Simple Machines


Simple Machines• Simple machines alter the force

and the distance moved.• For the inclined plane shown:

– F2 < F1 so MA >1 and d2 > d1

• If the ramp is frictionless, the work is the same in both cases.– F1d1 = F2d2

• With friction, F2d2 > F1d1.– The force is reduced but the work

done is greater.


Efficiency of Simple Machines

• Efficiency measures work output compared to work input.– In the absence of friction, they are equal.

• Real machines always have efficiencies less than 1, but they make work easier by changing the force required to do the work.

eff Wout

Win


Preview

• Multiple Choice

• Short Response

• Extended Response


Multiple Choice1. An object moves in a circle at a constant speed.

Which of the following is not true of the object?

A. Its acceleration is constant.

B. Its tangential speed is constant.

C. Its velocity is constant.

D. A centripetal force acts on the object.


Multiple Choice1. An object moves in a circle at a constant speed.

Which of the following is not true of the object?

A. Its acceleration is constant.

B. Its tangential speed is constant.

C. Its velocity is constant.

D. A centripetal force acts on the object.


Use the passage below to answer questions 2–3.

A car traveling at 15 m/s on a flat surface turns in a circle with a radius of 25 m.

2. What is the centripetal acceleration of the car?

F. 2.4 10-2 m/s2

G. 0.60 m/s2

H. 9.0 m/s2

J. zero

Multiple Choice, continued


Multiple Choice, continuedUse the passage below to answer questions 2–3.


2. What is the centripetal acceleration of the car?

F. 2.4 10-2 m/s2

G. 0.60 m/s2

H. 9.0 m/s2

J. zero




3. What is the most direct cause of the car’s centripetal acceleration?

A. the torque on the steering wheel

B. the torque on the tires of the car

C. the force of friction between the tires and the road

D. the normal force between the tires and the road




3. What is the most direct cause of the car’s centripetal acceleration?

A. the torque on the steering wheel

B. the torque on the tires of the car

C. the force of friction between the tires and the road

D. the normal force between the tires and the road


Multiple Choice, continued4. Earth (m = 5.97 1024 kg) orbits the sun (m =

1.99 1030 kg) at a mean distance of 1.50 1011 m. What is the gravitational force of the sun on Earth? (G = 6.673 10-11 N•m2/kg2)

F. 5.29 1032 N

G. 3.52 1022 N

H. 5.90 10–2 N

J. 1.77 10–8 N


Multiple Choice, continued4. Earth (m = 5.97 1024 kg) orbits the sun (m =

1.99 1030 kg) at a mean distance of 1.50 1011 m. What is the gravitational force of the sun on Earth? (G = 6.673 10-11 N•m2/kg2)

F. 5.29 1032 N

G. 3.52 1022 N

H. 5.90 10–2 N

J. 1.77 10–8 N



5. Which of the following is a correct interpretation of

the expression ?

A. Gravitational field strength changes with an object’s distance from Earth.

B. Free-fall acceleration changes with an object’s distance from Earth.

C. Free-fall acceleration is independent of the falling object’s mass.

D. All of the above are correct interpretations.

ag g GmEr2



5. Which of the following is a correct interpretation of

the expression ?

A. Gravitational field strength changes with an object’s distance from Earth.

B. Free-fall acceleration changes with an object’s distance from Earth.

C. Free-fall acceleration is independent of the falling object’s mass.

D. All of the above are correct interpretations.

ag g GmEr2


Multiple Choice, continued6. What data do you need to calculate the orbital

speed of a satellite?

F. mass of satellite, mass of planet, radius of orbit

G. mass of satellite, radius of planet, area of orbit

H. mass of satellite and radius of orbit only

J. mass of planet and radius of orbit only


Multiple Choice, continued6. What data do you need to calculate the orbital

speed of a satellite?

F. mass of satellite, mass of planet, radius of orbit

G. mass of satellite, radius of planet, area of orbit

H. mass of satellite and radius of orbit only

J. mass of planet and radius of orbit only


Multiple Choice, continued7. Which of the following choices correctly describes

the orbital relationship between Earth and the sun?

A. The sun orbits Earth in a perfect circle.

B. Earth orbits the sun in a perfect circle.

C. The sun orbits Earth in an ellipse, with Earth

at one focus.

D. Earth orbits the sun in an ellipse, with the sun

at one focus.


Multiple Choice, continued7. Which of the following choices correctly describes

the orbital relationship between Earth and the sun?

A. The sun orbits Earth in a perfect circle.

B. Earth orbits the sun in a perfect circle.

C. The sun orbits Earth in an ellipse, with Earth

at one focus.

D. Earth orbits the sun in an ellipse, with the sun

at one focus.



Use the diagram to answer

questions 8–9.8. The three forces acting onthe wheel have equal magnitudes. Which force willproduce the greatest torque on the wheel?

F. F1

G. F2

H. F3

J. Each force will produce the same torque.




questions 8–9.8. The three forces acting onthe wheel have equal magnitudes. Which force willproduce the greatest torque on the wheel?

F. F1

G. F2

H. F3

J. Each force will produce the same torque.




questions 8–9.9. If each force is 6.0 N, the angle between F1 and F2 is 60.0°, and the radius of the wheel is 1.0 m, what is theresultant torque on the wheel?

A. –18 N•m C. 9.0 N•m

B. –9.0 N•m D. 18 N•m




questions 8–9.9. If each force is 6.0 N, the angle between F1 and F2 is 60.0°, and the radius of the wheel is 1.0 m, what is theresultant torque on the wheel?

A. –18 N•m C. 9.0 N•m

B. –9.0 N•m D. 18 N•m


Multiple Choice, continued10. A force of 75 N is applied to a lever. This force

lifts a load weighing 225 N. What is the mechanical advantage of the lever?

F. 1/3

G. 3

H. 150

J. 300


Multiple Choice, continued10. A force of 75 N is applied to a lever. This force

lifts a load weighing 225 N. What is the mechanical advantage of the lever?

F. 1/3

G. 3

H. 150

J. 300


Multiple Choice, continued11. A pulley system has an efficiency of 87.5

percent. How much work must you do to lift a desk weighing 1320 N to a height of 1.50 m?

A. 1510 J

B. 1730 J

C. 1980 J

D. 2260 J


Multiple Choice, continued11. A pulley system has an efficiency of 87.5

percent. How much work must you do to lift a desk weighing 1320 N to a height of 1.50 m?

A. 1510 J

B. 1730 J

C. 1980 J

D. 2260 J


Multiple Choice, continued12. Which of the following statements is correct?

F. Mass and weight both vary with location.

G. Mass varies with location, but weight does not.

H. Weight varies with location, but mass does

not.

J. Neither mass nor weight varies with location.


Multiple Choice, continued12. Which of the following statements is correct?

F. Mass and weight both vary with location.

G. Mass varies with location, but weight does not.

H. Weight varies with location, but mass does

not.

J. Neither mass nor weight varies with location.


Multiple Choice, continued13. Which astronomer discovered that planets

travel in elliptical rather than circular orbits?

A. Johannes Kepler

B. Nicolaus Copernicus

C. Tycho Brahe

D. Claudius Ptolemy


Multiple Choice, continued13. Which astronomer discovered that planets

travel in elliptical rather than circular orbits?

A. Johannes Kepler

B. Nicolaus Copernicus

C. Tycho Brahe

D. Claudius Ptolemy


Short Response14. Explain how it is possible for all the water to

remain in a pail that is whirled in a vertical path, as shown below.


Short Response14. Explain how it is possible for all the water to

remain in a pail that is whirled in a vertical path, as shown below.

Answer: The water remains in the pail even when the pail is upside down because the water tends to move in a straight path due to inertia.


Short Response, continued15. Explain why approximately two high tides take

place every day at a given location on Earth.


Short Response, continued15. Explain why approximately two high tides take

place every day at a given location on Earth.

Answer: The moon’s tidal forces create two bulges on Earth. As Earth rotates on its axis once per day, any given point on Earth passes through both bulges.


Short Response, continued16. If you used a machine to increase the output

force, what factor would have to be sacrificed? Give an example.


Short Response, continued16. If you used a machine to increase the output

force, what factor would have to be sacrificed? Give an example.

Answer: You would have to apply the input force over a greater distance. Examples may include any machines that increase output force at the expense of input distance.


Extended Response

17. Mars orbits the sun (m = 1.99 1030 kg) at a mean distance of 2.28 1011 m. Calculate the length of

the Martian year in Earth days. Show all of your work. (G = 6.673 10–11 N•m2/kg2)


Extended Response

17. Mars orbits the sun (m = 1.99 1030 kg) at a mean distance of 2.28 1011 m. Calculate the length of

the Martian year in Earth days. Show all of your work. (G = 6.673 10–11 N•m2/kg2)

Answer: 687 days

Circular Motion and GravitationSection 1 Preview Section 1 Circular MotionCircular Motion Section 2 Newton’s Law of Universal GravitationNewton’s Law of.

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