Preview Section 1 Circular Motion

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? When asking students to express their ideas, you might try one of the following methods. (1) You could ask them to write their answers in their notebook and then discuss them. (2) You could ask them to first write their ideas and then share them with a small group of 3 or 4 students. At that time you can have each group present their consensus idea. This can be facilitated with the use of whiteboards for the groups. The most important aspect of eliciting student’s ideas is the acceptance of all ideas as valid. Do not correct or judge them. You might want to ask questions to help clarify their answers. You do not want to discourage students from thinking about these questions and just waiting for the correct answer from the teacher. Thank them for sharing their ideas. Misconceptions are common and can be dealt with if they are first expressed in writing and orally. In this example, students should be encouraged to make sketches and consider each situation. It is possible that responses will vary greatly. Some may believe that the force on the ball is inward, while others believe it is outward. Their explanations of these ideas are important and will help them focus on their ideas regarding motion in a circular path. Some may not believe a force is required, and there will likely be many who believe the force is toward the center in some cases and away from the center in other cases. Take the time to sort out the class ideas regarding this question before continuing. The idea of “centrifugal force” will be discussed later, so just encourage them to express their ideas.

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? The horse near the edge has the greater speed because it must travel a greater distance (2r) in the same amount of time. Tangential speed is simply the circumference divided by the period (time required to complete one revolution).

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” Point to the initial and final velocity vectors in the top diagram. Write the equation a= v/t, and point out that acceleration is the same direction as v. Since v = vf - vi or vf + (-vi), the lower diagram shows the value and direction for v. Emphasize that centripetal acceleration is always directed toward the center of the circle.

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? Perform the following demonstration before introducing the centripetal acceleration equation. You will need a soft object, like a rubber stopper, securely fastened to some string. Do not use a hard or heavy object. Spin the rubber stopper around in a circle at a uniform rate, say once every two seconds. Spin it faster, once every second, and ask the students if the acceleration would be greater or less. You might refer back to the diagram on the previous slide, and ask if the change in velocity per second is greater or less when moving at a high speed. Increasing the speed increases the acceleration as the square of the speed. Shorten the string by letting it wrap around your finger, and ask them how this would affect the acceleration. A smaller radius requires a greater acceleration to maintain the circular motion, so acceleration and radius are inversely related.

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 You can demonstrate tangential acceleration with the rubber stopper (see notes on the previous slide) by making it go faster or slower as you maintain the same radius for the circle.

Centripetal Acceleration
Click below to watch the Visual Concept. Visual Concept

Centripetal Force (Fc)
Remind students that accelerations only occur if a net force exists (F=ma). Then walk through the derivation for Fc. Return to the rubber stopper and swing it around (see notes on slide 4). Ask students to describe the force on the stopper from the string. It is in toward the center and slightly upward (to balance the force of gravity on the stopper). Overall, the net force is toward the center, just like the acceleration. A good simulation of the affect changing the speed has on the force can be found at: Choose “dynamics,” then choose “Circular Motion and Centripetal Force”. Follow the directions at the bottom of the screen to demonstrate Fc to the students. NOTE: These files can be downloaded for offline use. You must register with the site first and then the files can be sent to your address.

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 Before showing the diagram, ask students to answer the question. The answers may vary because there are misconceptions about inertia and “centrifugal force”, You could release the rubber stopper or better yet, have students spin a small sphere around on a lab table and release it at different points to see it moves TANGENT TO THE CIRCLE in a straight line. You may have students that expect it to continue in a curved path but it will actually travel in a straight line tangent to the circle (Newton’s 1st Law). Others may expect it to fly outward away from the center.

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. This topic requires some discussion with the students. Allow them to answer the two questions first. Ask them how it would be different if the car turned slowly around the corner. What keeps them from hitting the door in this case? The answer is the friction between the them and the seat. At high speed,s the friction is not great enough, so they continue in a relatively straight line while the door next to them turns to the left. In a fairly short amount of time, the door hits them. You can show this just by holding an object to represent the door and another object to represent the passenger. Move them in a straight line then turn the door to the left while the passenger goes straight. The text will not use the term “centrifugal force”. Note: Students will sometimes refer to inertia as a force. It is not a force, but is simply a measure of your resistance to a change in velocity.

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 V = d/t = circumerfence/time = (2)(3.14)(2.50 m)/(2.25s) = 6.98 m/s a = v2/r = (6.98 m/s)2/(2.50m)= 19.5 m/s2 It is worth pointing out to students that this is nearly double the acceleration due to gravity and, therefore, the force toward the center of the merry-go-round will be twice her weight. F = ma = (35.0 kg)(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? An all cases, a force is required. If there were no net force, the objects would not travel in a circle but instead would move in a straight line at a constant speed. In all cases, the force is toward the center of the circle. Source of the force on each object: car: friction between tires and road ball: the string moon: gravitational field of Earth child: friction between shoes and merry-go-round (or the bar if she is holding on) Point out that the forces all have different sources. Centripetal force is really any force that causes motion in a circle.

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? When asking students to express their ideas, you might try one of the following methods. (1) You could ask them to write their answers in their notebook and then discuss them. (2) You could ask them to first write their ideas and then share them with a small group of 3 or 4 students. At that time you can have each group present their consensus idea. This can be facilitated with the use of whiteboards for the groups. The most important aspect of eliciting student’s ideas is the acceptance of all ideas as valid. Do not correct or judge them. You might want to ask questions to help clarify their answers. You do not want to discourage students from thinking about these questions and just waiting for the correct answer from the teacher. Thank them for sharing their ideas. Misconceptions are common and can be dealt with if they are first expressed in writing and orally. Students often say “gravity” is the source. Do not settle for this answer. Ask them what they mean by gravity. The follow-up questions are included to see if students’ ideas include air pressure as a source of the force of gravity, or if they believe the spinning motion of Earth causes gravity. It is important to get them to write or verbalize their beliefs about gravity so they can move beyond statements such as “Gravity makes objects fall.”

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. See if students can draw the same conclusion as Newton before showing the result (second and third bullet points). You might draw Earth and place a cannon on a mountaintop. First show the path for very low speeds, and then show them a speed that produces a circular orbit. The picture in the slide shows the earth’s curvature if you look at the horizon. The curvature matches that of the gold-colored path. (In fact, this will not occur because of air friction. Discuss this issue if students raise it.) Students may ask what speed is required for orbit. This would be a nice problem for them to solve when they learn Kepler’s Laws in the next section.

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 = x N•m2/kg2 Point out that this law is universal. It does not simply apply to large masses. It applies to all masses

Law of Universal Gravitation
Point out that the forces are equal (same length), in accordance with Newton’s third law of motion. The force on the moon causes it to orbit, and the force on Earth causes it to orbit. They orbit around a common center of mass. The moon does not orbit around the center of Earth. The center of mass is inside Earth because Earth is much more massive than the moon. So although it may seem like the moon goes around the center of Earth, they actually both orbit a point between them (but much closer to Earth’s center).

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. You might show them the equation for Fg in the following form: Fg = ( N•m2/kg2 )(m1m2/r2) They will see that unless either m1 or m2 or the product of the two is a large number, the force will be very small.

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? If students just say gravity, ask why it would make the water rise. Spend some time finding out what students know about the tides. Doesn’t gravity pull up on Earth as well? If students know that there are two tides each day, ask them why there would be a high tide on the side opposite the moon if it is the moon’s gravitational pull that is causing the tides. This may create some confusion prior to the explanation. There are many excellent web sites with explanations and drawings showing the cause of the tides. These might be helpful in explaining the tides. The sun produces tides also, but of less significance. Because the sun is so far away, the difference between the near side of Earth and the center of Earth is proportionally much less than the distances measured from the moon.

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 The actual high tide is a little while after the moon is directly overhead. Many students may believe that the moon is only overhead at night. Since they don’t see it at 11:00 AM, it is a reasonable conclusion, but it is incorrect.

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. Ask students what each arrow represents. (The gravitational force acting on a unit mass at that location.) Then ask why the arrows in the outer circle are smallest. (They are smallest because gravitational force is inversely proportional to distance squared, so the force decreases as distance from Earth’s center increases.)

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. Help students understand that g and ag have the same value but are used for different situations. g is the force acting due to gravity (per unit mass). We use this to find the force on objects at rest, such as a book on your desk. ag is used to calculate the acceleration of falling objects. Also emphasize that the equation on the slide is for the value of g on Earth. Students can find the value on the surface of another planet by using the mass and radius of that planet.

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: N/m or m/s2 Help students solve these problems. There may be calculator issues to deal with regarding scientific notation and order of operations. This provides a good opportunity to help students get a ballpark answer by using orders of magnitude. Point out to the students that, when finding the value for g, they are also finding the value for ag. The second problem can be solved by using the answer to the first problem and dividing by the mass of the moon. However, it is a good idea to show them both methods. It will reinforce the derivation of the equation for g. Ask students what the answer to the second problem signifies. (The moon, or any other object at that distance from Earth, would accelerate toward Earth at a rate of m/s2 .) How would this acceleration change as the object moves closer to Earth? (The acceleration would increase.)

Now 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? The source is the gravitational field produced by Earth, or the attraction between Earth and the mass. It could only be diminished by moving it farther from the center of Earth. Placing it in a vacuum would have little effect. There is a very slight buoyant force from the air that is displaced, but this is not significant. (The point of this question is to dispel the myth that air pressure somehow causes the downward force on objects.) Earth’s rotation is also of little significance. There is a slight adjustment in weight because of the centripetal force necessary to maintain circular motion, so the weight would increase very slightly if Earth stopped spinning, but not because gravity would change in any way.

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. When asking students to express their ideas, you might try one of the following methods. (1) You could ask them to write their answers in their notebook and then discuss them. (2) You could ask them to first write their ideas and then share them with a small group of 3 or 4 students. At that time you can have each group present their consensus idea. This can be facilitated with the use of whiteboards for the groups. The most important aspect of eliciting student’s ideas is the acceptance of all ideas as valid. Do not correct or judge them. You might want to ask questions to help clarify their answers. You do not want to discourage students from thinking about these questions and just waiting for the correct answer from the teacher. Thank them for sharing their ideas. Misconceptions are common and can be dealt with if they are first expressed in writing and orally. Listen to students’ ideas and try to get clarification for ideas that are vague. Is the path drawn circular? elliptic? something else? Get them to commit to an answer. Does Earth travel at a constant speed? If not, how does the speed change? How do the other planets compare? Other than distance from the sun, are there other differences?

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? When asking students to express their ideas, you might try one of the following methods. (1) You could ask them to write their answers in their notebook and then discuss them. (2) You could ask them to first write their ideas and then share them with a small group of 3 or 4 students. At that time you can have each group present their consensus idea. This can be facilitated with the use of whiteboards for the groups. The most important aspect of eliciting student’s ideas is the acceptance of all ideas as valid. Do not correct or judge them. You might want to ask questions to help clarify their answers. You do not want to discourage students from thinking about these questions and just waiting for the correct answer from the teacher. Thank them for sharing their ideas. Misconceptions are common and can be dealt with if they are first expressed in writing and orally. Most students will probably refer to the astronauts and the videos they have seen of them over the years. Some may know of the plane used to practice being weightless. Some may say in outer space you are weightless. You might ask them “where” in outer space you need to be for this to occur.

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. Remind students that the scale reading is the normal force. In the second case, when an elevator starts to move downward, you feel “lighter” for a brief moment. (After this, the elevator returns to a constant speed, and the scale reading goes back up to its initial value.) Ask students what would happen if the person in the free-falling elevator held an apple out in front of his face and let go. (The apple would remain in the same spot because it is falling with the same acceleration as him, 9.81 m/s2.) Students may have had similar experiences on amusement park free-fall rides.

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. Remind students that orbiting means falling at the same rate that Earth curves away from you (while moving sideways), so you never get any closer.

Now 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. The paths should be slightly elliptical. Some planets are more elliptical than others. The planet moves fastest when nearest the sun and slower at the greatest distance. (The reason for this can be seen from the web site recommended for Kepler’s laws. The force does not act perpendicular to the velocity. It has a component causing acceleration and deceleration at various point on the ellipse.)

Now 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? Most weightlessness is apparent. No support force is required. (Everyone can experience apparent weightlessness for a brief period of time. Just jump off a table and no support force is required for about 0.4 seconds.)

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.

Overview of Simple Machines
Click below to watch the Visual Concept. Visual Concept

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. The diagram depicts the task of lifting a heavy load into the back of a truck. If friction is disregarded, the work is the same either way because the ramp reduces the force and increases the distance by the same factor. In reality, on the ramp the force must also overcome friction, so it is actually more work. However, the work is easier to do because the applied force is less.

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.

Preview Multiple Choice Short Response Extended Response

Multiple Choice 1. 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 Choice, continued
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 m/s2 H. 9.0 m/s2 J. zero

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. 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

4. Earth (m = 5.97  1024 kg) orbits the sun (m =  1030 kg) at a mean distance of 1.50  1011 m. What is the gravitational force of the sun on Earth? (G =  N•m2/kg2) F  1032 N G  1022 N H  10–2 N J  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.

6. 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

7. 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

Use the diagram to answer questions 8–9. 8. The three forces acting on the wheel have equal magnitudes. Which force will produce the greatest torque on the wheel? F. F1 G. F2 H. F3 J. Each force will produce the same torque.

Use the diagram to answer 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 the resultant torque on the wheel? A. –18 N•m C. 9.0 N•m B. –9.0 N•m D. 18 N•m

10. 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

11. A pulley system has an efficiency of 87.5 percent. How much work must you do to lift a desk weighing N to a height of 1.50 m? A J B J C J D J

12. 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.

13. 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 Response 14. 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 Response 14. 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, continued
15. Explain why approximately two high tides take place every day at a given location on Earth.

15. 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.

16. If you used a machine to increase the output force, what factor would have to be sacrificed? Give an example.

16. 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 =  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 =  10–11 N•m2/kg2) Answer: 687 days

Preview Section 1 Circular Motion

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