1. College Physics, 7th Edition
Lecture Outline
Chapter 7
College Physics, 7th Edition
Wilson / Buffa / Lou
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2. Chapter 7 Circular Motion and Gravitation
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3. Units of Chapter 7 Angular Measure Angular Speed and Velocity
Uniform Circular Motion and Centripetal Acceleration
Angular Acceleration
Newton’s Law of Gravitation
Kepler’s Laws and Earth Satellites
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4. 7.1 Angular Measure
The position of an object can be described using polar coordinates—r and θ—rather than x and y. The figure at left gives the conversion between the two descriptions.
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5. 7.1 Angular Measure
It is most convenient to measure the angle θ in radians:
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6. 7.1 Angular Measure
The small-angle approximation is very useful, as it allows the substitution of θ for sin θ when the angle is sufficiently small.
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7. 7.2 Angular Speed and Velocity
In analogy to the linear case, we define the average and instantaneous angular speed:
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8. 7.2 Angular Speed and Velocity
The direction of the angular velocity is along the axis of rotation, and is given by a right-hand rule.
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9. 7.2 Angular Speed and Velocity
Relationship between tangential and angular speeds:
This means that parts of a rotating object farther from the axis of rotation move faster.
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10. 7.2 Angular Speed and Velocity
The period is the time it takes for one rotation; the frequency is the number of rotations per second.
The relation of the frequency to the angular speed:
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11. 7.3 Uniform Circular Motion and Centripetal Acceleration
A careful look at the change in the velocity vector of an object moving in a circle at constant speed shows that the acceleration is toward the center of the circle.
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12. 7.3 Uniform Circular Motion and Centripetal Acceleration
The same analysis shows that the centripetal acceleration is given by:
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13. 7.3 Uniform Circular Motion and Centripetal Acceleration
The centripetal force is the mass multiplied by the centripetal acceleration.
This force is the net force on the object. As the force is always perpendicular to the velocity, it does no work.
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14. 7.4 Angular Acceleration
The average angular acceleration is the rate at which the angular speed changes:
In analogy to constant linear acceleration:
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15. 7.4 Angular Acceleration
If the angular speed is changing, the linear speed must be changing as well. The tangential acceleration is related to the angular acceleration:
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16. 7.4 Angular Acceleration
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17. 7.5 Newton’s Law of Gravitation
Newton’s law of universal gravitation describes the force between any two point masses:
G is called the universal gravitational constant:
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18. 7.5 Newton’s Law of Gravitation
Gravity provides the centripetal force that keeps planets, moons, and satellites in their orbits.
We can relate the universal gravitational force to the local acceleration of gravity:
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19. 7.5 Newton’s Law of Gravitation
The gravitational potential energy is given by the general expression:
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20. 7.6 Kepler’s Laws and Earth Satellites
Kepler’s laws were the result of his many years of observations. They were later found to be consequences of Newton’s laws.
Kepler’s first law:
Planets move in elliptical orbits, with the Sun at one of the focal points.
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21. 7.6 Kepler’s Laws and Earth Satellites
Kepler’s second law:
A line from the Sun to a planet sweeps out equal areas in equal lengths of time.
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22. 7.6 Kepler’s Laws and Earth Satellites
Kepler’s third law:
The square of the orbital period of a planet is directly proportional to the cube of the average distance of the planet from the Sun; that is,
This can be derived from Newton’s law of gravitation, using a circular orbit.
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23. 7.6 Kepler’s Laws and Earth Satellites
If a projectile is given enough speed to just reach the top of the Earth’s gravitational well, its potential energy at the top will be zero. At the minimum, its kinetic energy will be zero there as well.
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24. 7.6 Kepler’s Laws and Earth Satellites
This minimum initial speed is called the escape speed.
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25. 7.6 Kepler’s Laws and Earth Satellites
Any satellite in orbit around the Earth has a speed given by
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26. 7.6 Kepler’s Laws and Earth Satellites
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27. 7.6 Kepler’s Laws and Earth Satellites
Astronauts in Earth orbit report the sensation of weightlessness. The gravitational force on them is not zero; what’s happening?
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28. 7.6 Kepler’s Laws and Earth Satellites
What’s missing is not the weight, but the normal force. We call this apparent weightlessness.
“Artificial” gravity could be produced in orbit by rotating the satellite; the centripetal force would mimic the effects of gravity.
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29. Summary of Chapter 7
Angles may be measured in radians; the angle is the arc length divided by the radius.
Angular kinematic equations for constant acceleration:
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30. Summary of Chapter 7
Tangential speed is proportional to angular speed.
Frequency is inversely proportional to period.
Angular speed:
Centripetal acceleration:
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31. Summary of Chapter 7 Centripetal force:
Angular acceleration is the rate at which the angular speed changes. It is related to the tangential acceleration.
Newton’s law of gravitation:
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32. Summary of Chapter 7 Gravitational potential energy: Kepler’s laws:
Planetary orbits are ellipses with Sun at one focus
Equal areas are swept out in equal times.
The square of the period is proportional to the cube of the radius.
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33. Summary of Chapter 7 Escape speed from Earth:
Energy of a satellite orbiting Earth:
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