Chapter 7 Rotational Motion and The Law of Gravity

1. Chapter 7 Rotational Motion and The Law of Gravity

2. The radian • The radian is a unit of angular measure • The angle in radians can be defined as the ratio of the arc length s along a circle divided by the radius r

3. Rotation of a rigid body • We consider rotational motion of a rigid body about a fixed axis • Rigid body rotates with all its parts locked together and without any change in its shape • Fixed axis: it does not move during the rotation • This axis is called axis of rotation • Reference line is introduced

4. Angular position • Reference line is fixed in the body, is perpendicular to the rotation axis, intersects the rotation axis, and rotates with the body • Angular position – the angle (in radians or degrees) of the reference line relative to a fixed direction (zero angular position)

5. Angular displacement • Angular displacement – the change in angular position. • Angular displacement is considered positive in the CCW direction and holds for the rigid body as a whole and every part within that body

6. Angular velocity • Average angular velocity • Instantaneous angular velocity – the rate of change in angular position

7. Angular acceleration • Average angular acceleration • Instantaneous angular acceleration – the rate of change in angular velocity

8. Uniform circular motion • A special case of 2D motion • An object moves around a circle at a constant speed • Period – time to make one full revolution • An object traveling in a circle, even though it moves with a constant speed, will have an acceleration

9. Centripetal acceleration • Centripetal acceleration is due to the change in the direction of the velocity • Centripetal acceleration is directed toward the center of the circle of motion

10. Centripetal acceleration • The magnitude of the centripetal acceleration is given by

11. Centripetal acceleration • During a uniform circular motion: • the speed is constant • the velocity is changing due to centripetal(“center seeking”) acceleration • centripetal acceleration is constant in magnitude (v2/r), is normal to the velocity vector, and points radially inward

12. Rotation with constant angular acceleration • Similarly to the case of 1D motion with a constant acceleration we can derive a set of formulas:

13. Chapter 7 Problem 5 A dentist’s drill starts from rest. After 3.20 s of constant angular acceleration, it turns at a rate of 2.51 × 104 rev/min. (a) Find the drill’s angular acceleration. (b) Determine the angle (in radians) through which the drill rotates during this period.

14. Relating the linear and angular variables: position • For a point on a reference line at a distance r from the rotation axis: • θis measured in radians

15. Relating the linear and angular variables: speed • ωis measured in rad/s • Period

16. Chapter 7 Problem 2 A wheel has a radius of 4.1 m. How far (path length) does a point on the circumference travel if the wheel is rotated through angles of 30°, 30 rad, and 30 rev, respectively?

17. Relating the linear and angular variables: acceleration • αis measured in rad/s2 • Centripetal acceleration

18. Total acceleration • Tangential acceleration is due to changing speed • Centripetal acceleration is due to changing direction • Total acceleration:

19. Centripetal force • For an object in a uniform circular motion, the centripetal acceleration is • According to the Newton’s Second Law, a force must cause this acceleration – centripetal force • A centripetal force accelerates a body by changing the direction of the body’s velocity without changing the speed

20. Centripetal force • Centripetal forces may have different origins • Gravitation can be a centripetal force • Tension can be a centripetal force • Etc.

21. Centripetal force • Centripetal forces may have different origins • Gravitation can be a centripetal force • Tension can be a centripetal force • Etc.

22. Free-body diagram

23. Chapter 7 Problem 28 A roller-coaster vehicle has a mass of 500 kg when fully loaded with passengers. (a) If the vehicle has a speed of 20.0 m/s at point A, what is the force exerted by the track on the car at this point? (b) What is the in maximum speed the vehicle can have at point B and still remain on the track?

24. Newton’s law of gravitation • Any two (or more) massive bodies attract each other • Gravitational force (Newton's law of gravitation) • Gravitational constantG= 6.67*10 –11 N*m2/kg2 = 6.67*10 –11 m3/(kg*s2) – universal constant

25. Gravitation and the superposition principle • For a group of interacting particles, the net gravitational force on one of the particles is

26. Chapter 7 Problem 33 Objects with masses of 200 kg and 500 kg are separated by 0.400 m. (a) Find the net gravitational force exerted by these objects on a 50.0-kg object placed midway between them. (b) At what position (other than infinitely remote ones) can the 50.0-kg object be placed so as to experience a net force of zero?

27. Gravity force near the surface of Earth • Earth can be though of as a nest of shells, one within another and each attracting a particle outside the Earth’s surface • Thus Earth behaves like a particle located at the center of Earth with a mass equal to that of Earth • g = 9.8 m/s2 • This formula is derived for stationary Earth of ideal spherical shape and uniform density

28. Gravity force near the surface of Earth In reality gis not a constant because: Earth is rotating, Earth is approximately an ellipsoid with a non-uniform density

29. Gravitational potential energy • Gravitation is a conservative force (work done by it is path-independent) • For conservative forces potential energy can be introduced • Gravitational potential energy:

30. Gravitational potential energy

31. Escape speed • Accounting for the shape of Earth, projectile motion (Ch. 3) has to be modified:

32. Escape speed • Escape speed: speed required for a particle to escape from the planet into infinity (and stop there)

33. Escape speed • If for some astronomical object • Nothing (even light) can escape from the surface of this object – a black hole

34. Escape speed

35. Chapter 7 Problem 56 Show that the escape speed from the surface of a planet of uniform density is directly proportional to the radius of the planet.

36. Johannes Kepler (1571-1630) Tycho Brahe/ Tyge Ottesen Brahe de Knudstrup (1546-1601) • Kepler’s laws • Three Kepler’s laws • 1. The law of orbits: All planets move in elliptical orbits, with the Sun at one focus • 2. The law of areas: A line that connects the planet to the Sun sweeps out equal areas in the plane of the planet’s orbit in equal time intervals • 3. The law of periods: The square of the period of any planet is proportional to the cube of the semimajor axis of its orbit

37. First Kepler’s law • All planets move in elliptical orbits with the Sun at one focus, whereas the second focus is empty • Any object bound to another by an inverse square law will move in an elliptical path

38. Second Kepler’s law • A line drawn from the Sun to any planet will sweep out equal areas in equal times • Area from A to B and C to D are the same

39. Third Kepler’s law • For a circular orbit and the Newton’s Second law • From the definition of a period

40. Satellites • For a circular orbit and the Newton’s Second law • Kinetic energy of a satellite • Total mechanical energy of a satellite

41. Chapter 7 Problem 45 The Solar Maximum Mission Satellite was placed in a circular orbit about 150 mi above Earth. Determine (a) the orbital speed of the satellite and (b) the time required for one complete revolution.

42. Questions?

43. Answers to the even-numbered problems Chapter 7 Problem 10 50 rev

44. Answers to the even-numbered problems • Chapter 7 • Problem 30 • 4.39 × 1020 N toward the Sun • (b) 1.99 × 1020 N toward the Sun • (c) 3.55 × 1020 N toward the Sun

45. Answers to the even-numbered problems • Chapter 7 • Problem 36 • 5.59 × 103 m/s • (b) 3.98 h • (c) 1.47 × 103 N