Physics Pp Presentation Ch 7

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Chapter Presentation Transparencies Sample Problems Visual Concepts Standardized Test Prep Resources

Table of Contents Section 1 Circular Motion Section 2 Newton’s Law of Universal Gravitation Section 3 Motion in Space Section 4 Torque and Simple Machines Circular Motion and Gravitation Chapter 7

Objectives Solve problems involving centripetal acceleration. Solve problems involving centripetal force. Explain how the apparent existence of an outward force in circular motion can be explained as inertia resisting the centripetal force. Chapter 7 Section 1 Circular Motion

Tangential Speed The tangential speed ( v t ) of an object in circular motion is the object’s speed along an imaginary line drawn tangent to the circular path. Tangential speed depends on the distance from the object to the center of the circular path. When the tangential speed is constant, the motion is described as uniform circular motion. Chapter 7 Section 1 Circular Motion

Centripetal Acceleration Chapter 7 Section 1 Circular Motion

Centripetal Acceleration The acceleration of an object moving in a circular path and at constant speed is due to a change in direction. An acceleration of this nature is called a centripetal acceleration. Chapter 7 Section 1 Circular Motion

Centripetal Acceleration, continued (a) As the particle moves from A to B , the direction of the particle’s velocity vector changes. (b) For short time intervals, ∆v is directed toward the center of the circle. Centripetal acceleration is always directed toward the center of a circle. Chapter 7 Section 1 Circular Motion

Centripetal Acceleration, continued You have seen that centripetal acceleration results from a change in direction. In circular motion, an acceleration due to a change in speed is called tangential acceleration. To understand the difference between centripetal and tangential acceleration, consider a car traveling in a circular track . Because the car is moving in a circle, the car has a centripetal component of acceleration. If the car’s speed changes, the car also has a tangential component of acceleration. Chapter 7 Section 1 Circular Motion

Centripetal Force Consider a ball of mass m that is being whirled in a horizontal circular path of radius r with constant speed. Chapter 7 The force exerted by the string has horizontal and vertical components. The vertical component is equal and opposite to the gravitational force . Thus, the horizontal component is the net force. This net force, which is is directed toward the center of the circle, is a centripetal force. Section 1 Circular Motion

Centripetal Force, continued Newton’s second law can be combined with the equation for centripetal acceleration to derive an equation for centripetal force: Chapter 7 Section 1 Circular Motion

Centripetal Force, continued Chapter 7 Centripetal force is simply the name given to the net force on an object in uniform circular motion . Any type of force or combination of forces can provide this net force. For example, friction between a race car’s tires and a circular track is a centripetal force that keeps the car in a circular path. As another example, gravitational force is a centripetal force that keeps the moon in its orbit. Section 1 Circular Motion

Centripetal Force, continued Chapter 7 If the centripetal force vanishes, the object stops moving in a circular path. A ball that is on the end of a string is whirled in a vertical circular path. If the string breaks at the position shown in (a), the ball will move vertically upward in free fall. If the string breaks at the top of the ball’s path, as in (b), the ball will move along a parabolic path. Section 1 Circular Motion

Describing a Rotating System Chapter 7 To better understand the motion of a rotating system, consider a car traveling at high speed and approaching an exit ramp that curves to the left. As the driver makes the sharp left turn, the passenger slides to the right and hits the door. What causes the passenger to move toward the door? Section 1 Circular Motion

Describing a Rotating System, continued Chapter 7 Section 1 Circular Motion As the car enters the ramp and travels along a curved path, the passenger, because of inertia, tends to move along the original straight path. If a sufficiently large centripetal force acts on the passenger, the person will move along the same curved path that the car does. The origin of the centripetal force is the force of friction between the passenger and the car seat. If this frictional force is not sufficient, the passenger slides across the seat as the car turns underneath.

Objectives Explain how Newton’s law of universal gravitation accounts for various phenomena, including satellite and planetary orbits, falling objects, and the tides. Apply Newton’s law of universal gravitation to solve problems. Section 2 Newton’s Law of Universal Gravitation Chapter 7

Gravitational Force Orbiting objects are in free fall. To see how this idea is true, we can use a thought experiment that Newton developed. Consider a cannon sitting on a high mountaintop. Chapter 7 Section 2 Newton’s Law of Universal Gravitation Each successive cannonball has a greater initial speed, so the horizontal distance that the ball travels increases. If the initial speed is great enough, the curvature of Earth will cause the cannonball to continue falling without ever landing.

Gravitational Force, continued The centripetal force that holds the planets in orbit is the same force that pulls an apple toward the ground— gravitational force. Gravitational force is the mutual force of attraction between particles of matter. Gravitational force depends on the masses and on the distance between them . Chapter 7 Section 2 Newton’s Law of Universal Gravitation

Gravitational Force, continued Newton developed the following equation to describe quantitatively the magnitude of the gravitational force if distance r separates masses m 1 and m 2 : Chapter 7 Section 2 Newton’s Law of Universal Gravitation The constant G , called the constant of universal gravitation , equals 6.673  10 –11 N•m 2 /kg.

Newton’s Law of Universal Gravitation Chapter 7 Section 2 Newton’s Law of Universal Gravitation

Gravitational Force, continued The gravitational forces that two masses exert on each other are always equal in magnitude and opposite in direction . This is an example of Newton’s third law of motion. One example is the Earth-moon system, shown on the next slide. As a result of these forces, the moon and Earth each orbit the center of mass of the Earth-moon system. Because Earth has a much greater mass than the moon, this center of mass lies within Earth. Chapter 7 Section 2 Newton’s Law of Universal Gravitation

Applying the Law of Gravitation Newton’s law of gravitation accounts for ocean tides. High and low tides are partly due to the gravitational force exerted on Earth by its moon. The tides result from the difference between the gravitational force at Earth’s surface and at Earth’s center. Chapter 7 Section 2 Newton’s Law of Universal Gravitation

Applying the Law of Gravitation, continued Cavendish applied Newton’s law of universal gravitation to find the value of G and Earth’s mass. When two masses , the distance between them, and the gravitational force are known, Newton’s law of universal gravitation can be used to find G . Once the value of G is known, the law can be used again to find Earth’s mass. Chapter 7 Section 2 Newton’s Law of Universal Gravitation

Gravity is a field force. Gravitational field strength, g , equals F g /m . The gravitational field, g , is a vector with magnitude g that points in the direction of F g . Gravitational field strength equals free-fall acceleration. Applying the Law of Gravitation, continued Chapter 7 Section 2 Newton’s Law of Universal Gravitation The gravitational field vectors represent Earth’s gravitational field at each point.

Applying the Law of Gravitation, continued weight = mass  gravitational field strength Because it depends on gravitational field strength , weight changes with location: Chapter 7 Section 2 Newton’s Law of Universal Gravitation On the surface of any planet, the value of g, as well as your weight, will depend on the planet’s mass and radius.

Objectives Describe Kepler’s laws of planetary motion. Relate Newton’s mathematical analysis of gravitational force to the elliptical planetary orbits proposed by Kepler. Solve problems involving orbital speed and period. Section 3 Motion in Space Chapter 7

Kepler’s Laws Kepler’s laws describe the motion of the planets. First Law: Each planet travels in an elliptical orbit around the sun, and the sun is at one of the focal points. Second Law: An imaginary line drawn from the sun to any planet sweeps out equal areas in equal time intervals. Third Law: The square of a planet’s orbital period ( T 2 ) is proportional to the cube of the average distance ( r 3 ) between the planet and the sun. Chapter 7 Section 3 Motion in Space

Kepler’s Laws, continued Kepler’s laws were developed a generation before Newton’s law of universal gravitation. Newton demonstrated that Kepler’s laws are consistent with the law of universal gravitation . The fact that Kepler’s laws closely matched observations gave additional support for Newton’s theory of gravitation. Chapter 7 Section 3 Motion in Space

Kepler’s Laws, continued According to Kepler’s second law, if the time a planet takes to travel the arc on the left (∆ t 1 ) is equal to the time the planet takes to cover the arc on the right (∆ t 2 ), then the area A 1 is equal to the area A 2 . Chapter 7 Thus, the planet travels faster when it is closer to the sun and slower when it is farther away. Section 3 Motion in Space

Kepler’s Laws, continued Kepler’s third law states that T 2  r 3 . The constant of proportionality is 4  2 / Gm , where m is the mass of the object being orbited. So, Kepler’s third law can also be stated as follows: Chapter 7 Section 3 Motion in Space

Kepler’s Laws, continued Kepler’s third law leads to an equation for the period of an object in a circular orbit. The speed of an object in a circular orbit depends on the same factors: Chapter 7 Note that m is the mass of the central object that is being orbited. The mass of the planet or satellite that is in orbit does not affect its speed or period. The mean radius ( r ) is the distance between the centers of the two bodies. Section 3 Motion in Space

Planetary Data Chapter 7 Section 3 Motion in Space

Sample Problem Period and Speed of an Orbiting Object Magellan was the first planetary spacecraft to be launched from a space shuttle. During the spacecraft’s fifth orbit around Venus, Magellan traveled at a mean altitude of 361km. If the orbit had been circular, what would Magellan’s period and speed have been? Chapter 7 Section 3 Motion in Space

Sample Problem, continued 1. Define Given: r 1 = 361 km = 3.61  10 5 m Unknown: T = ? v t = ? Chapter 7 2. Plan Choose an equation or situation: Use the equations for the period and speed of an object in a circular orbit. Section 3 Motion in Space

Sample Problem, continued Chapter 7 Use Table 1 in the textbook to find the values for the radius ( r 2 ) and mass ( m ) of Venus. r 2 = 6.05  10 6 m m = 4.87  10 24 kg Find r by adding the distance between the spacecraft and Venus’s surface ( r 1 ) to Venus’s radius ( r 2 ). r = r 1 + r 2 r = 3.61  10 5 m + 6.05  10 6 m = 6.41  10 6 m Section 3 Motion in Space

Sample Problem, continued Chapter 7 3. Calculate 4. Evaluate Magellan takes (5.66  10 3 s)(1 min/60 s)  94 min to complete one orbit. Section 3 Motion in Space

Weight and Weightlessness To learn about apparent weightlessness, imagine that you are in an elevator: When the elevator is at rest, the magnitude of the normal force acting on you equals your weight. If the elevator were to accelerate downward at 9.81 m/s 2 , you and the elevator would both be in free fall. You have the same weight, but there is no normal force acting on you. This situation is called apparent weightlessness. Astronauts in orbit experience apparent weightlessness. Chapter 7 Section 3 Motion in Space

Weight and Weightlessness Chapter 7 Section 3 Motion in Space

Objectives Distinguish between torque and force. Calculate the magnitude of a torque on an object. Identify the six types of simple machines. Calculate the mechanical advantage of a simple machine. Section 4 Torque and Simple Machines Chapter 7

Rotational Motion Rotational and translational motion can be analyzed separately. For example, when a bowling ball strikes the pins, the pins may spin in the air as they fly backward. These pins have both rotational and translational motion. In this section, we will isolate rotational motion. In particular, we will explore how to measure the ability of a force to rotate an object. Section 4 Torque and Simple Machines Chapter 7

The Magnitude of a Torque Torque is a quantity that measures the ability of a force to rotate an object around some axis. How easily an object rotates on both how much force is applied and on where the force is applied. The perpendicular distance from the axis of rotation to a line drawn along the direction of the force is equal to d sin  and is called the lever arm.  = Fd sin  torque = force  lever arm Section 4 Torque and Simple Machines Chapter 7

The Magnitude of a Torque, continued The applied force may act at an angle. However, the direction of the lever arm ( d sin  ) is always perpendicular to the direction of the applied force, as shown here. Section 4 Torque and Simple Machines Chapter 7

Torque Chapter 7 Section 4 Torque and Simple Machines

Torque and the Lever Arm Chapter 7 Section 4 Torque and Simple Machines In each example, the cat is pushing on the door at the same distance from the axis. To produce the same torque, the cat must apply greater force for smaller angles.

The Sign of a Torque Torque is a vector quantity. In this textbook, we will assign each torque a positive or negative sign, depending on the direction the force tends to rotate an object. We will use the convention that the sign of the torque is positive if the rotation is counterclockwise and negative if the rotation is clockwise. Section 4 Torque and Simple Machines Chapter 7 Tip: To determine the sign of a torque, imagine that the torque is the only one acting on the object and that the object is free to rotate. Visualize the direction that the object would rotate. If more than one force is acting, treat each force separately.

The Sign of a Torque Chapter 7 Section 4 Torque and Simple Machines

Sample Problem Torque A basketball is being pushed by two players during tip-off. One player exerts an upward force of 15 N at a perpendicular distance of 14 cm from the axis of rotation.The second player applies a downward force of 11 N at a distance of 7.0 cm from the axis of rotation. Find the net torque acting on the ball about its center of mass. Chapter 7 Section 4 Torque and Simple Machines

Sample Problem, continued 1. Define Given: F 1 = 15 N F 2 = 11 N d 1 = 0.14 m d 2 = 0.070 m Chapter 7 Section 4 Torque and Simple Machines Diagram: Unknown:  net = ?

Sample Problem, continued 2. Plan Choose an equation or situation: Apply the definition of torque to each force,and add up the individual torques. Chapter 7 Section 4 Torque and Simple Machines Tip: The factor sin  is not included in the torque equation because each given distance is the perpendicular distance from the axis of rotation to a line drawn along the direction of the force. In other words, each given distance is the lever arm.  = Fd  net =  1 +  2 = F 1 d 1 + F 2 d 2

Sample Problem, continued 4. Evaluate The net torque is negative,so the ball rotates in a clockwise direction. Chapter 7 Section 4 Torque and Simple Machines Calculate Substitute the values into the equation and solve: First,determine the torque produced by each force.Use the standard convention for signs.   1 = F 1 d 1 = (15 N)(–0.14 m) = –2.1 N•m  2 = F 2 d 2 = (–11 N)(0.070 m) = –0.77 N•m  net =  1 +  2 = –2.1 N•m – 0.77 N•m  net = –2.9 N•m

Simple Machines A machine is any device that transmits or modifies force, usually by changing the force applied to an object. All machines are combinations or modifications of six fundamental types of machines, called simple machines. These six simple machines are the lever, pulley, inclined plane, wheel and axle, wedge, and screw. Section 4 Torque and Simple Machines Chapter 7

Simple Machines Chapter 7 Section 4 Torque and Simple Machines

Simple Machines, continued Because the purpose of a simple machine is to change the direction or magnitude of an input force, a useful way of characterizing a simple machine is to compare the output and input force. This ratio is called mechanical advantage . If friction is disregarded , mechanical advantage can also be expressed in terms of input and output distance. Section 4 Torque and Simple Machines Chapter 7

Simple Machines, continued Section 4 Torque and Simple Machines Chapter 7 The diagrams show two examples of a trunk being loaded onto a truck. In the first example, a force ( F 1 ) of 360 N moves the trunk through a distance ( d 1 ) of 1.0 m. This requires 360 N•m of work. In the second example, a lesser force ( F 2 ) of only 120 N would be needed (ignoring friction), but the trunk must be pushed a greater distance ( d 2 ) of 3.0 m. This also requires 360 N•m of work.

Simple Machines, continued The simple machines we have considered so far are ideal, frictionless machines. Real machines, however, are not frictionless. Some of the input energy is dissipated as sound or heat. The efficiency of a machine is the ratio of useful work output to work input. Section 4 Torque and Simple Machines Chapter 7 The efficiency of an ideal (frictionless) machine is 1, or 100 percent. The efficiency of real machines is always less than 1.

Mechanical Efficiency Chapter 7 Section 4 Torque and Simple Machines

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. Standardized Test Prep Chapter 7

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/s 2 G. 0.60 m/s 2 H. 9.0 m/s 2 J. zero Standardized Test Prep Chapter 7

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. 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 Standardized Test Prep Chapter 7

Multiple Choice, continued 4. Earth ( m = 5.97  10 24 kg) orbits the sun ( m = 1.99  10 30 kg) at a mean distance of 1.50  10 11 m. What is the gravitational force of the sun on Earth? ( G = 6.673  10 -11 N•m 2 /kg 2 ) F. 5.29  10 32 N G. 3.52  10 22 N H. 5.90  10 –2 N J. 1.77  10 –8 N Standardized Test Prep Chapter 7

Multiple Choice, continued 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. Standardized Test Prep Chapter 7

Multiple Choice, continued 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 Standardized Test Prep Chapter 7

Multiple Choice, continued 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 at one focus. Standardized Test Prep Chapter 7

Multiple Choice, continued Use the diagram to answer questions 8–9. Standardized Test Prep Chapter 7 8. The three forces acting on the wheel have equal magnitudes. Which force will produce the greatest torque on the wheel? F. F 1 G. F 2 H. F 3 J. Each force will produce the same torque.

Multiple Choice, continued Use the diagram to answer questions 8–9. Standardized Test Prep Chapter 7 9. If each force is 6.0 N, the angle between F 1 and F 2 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

Multiple Choice, continued 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 Standardized Test Prep Chapter 7

Multiple Choice, continued 11. 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 Standardized Test Prep Chapter 7

Multiple Choice, continued 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. Standardized Test Prep Chapter 7

Multiple Choice, continued 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 Standardized Test Prep Chapter 7

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. Standardized Test Prep Chapter 7

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. Standardized Test Prep Chapter 7 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. Standardized Test Prep Chapter 7

Short Response, continued 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. Standardized Test Prep Chapter 7

Short Response, continued 16. If you used a machine to increase the output force, what factor would have to be sacrificed? Give an example. Standardized Test Prep Chapter 7

Short Response, continued 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. Standardized Test Prep Chapter 7

Extended Response 17. Mars orbits the sun ( m = 1.99  10 30 kg) at a mean distance of 2.28  10 11 m. Calculate the length of the Martian year in Earth days. Show all of your work. ( G = 6.673  10 –11 N•m 2 /kg 2 ) Standardized Test Prep Chapter 7

Extended Response 17. Mars orbits the sun ( m = 1.99  10 30 kg) at a mean distance of 2.28  10 11 m. Calculate the length of the Martian year in Earth days. Show all of your work. ( G = 6.673  10 –11 N•m 2 /kg 2 ) Standardized Test Prep Chapter 7 Answer: 687 days

Centripetal Force Chapter 7 Section 1 Circular Motion

Kepler’s Laws Chapter 7 Section 3 Motion in Space

The Magnitude of a Torque Section 4 Torque and Simple Machines Chapter 7

Simple Machines Section 4 Torque and Simple Machines Chapter 7

Physics Pp Presentation Ch 7

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