Circular Motion Dynamics: Analysis and Applications

1. Chapter 5 Dynamics of Uniform Circular Motion Chapter 8 Rotational Kinematics

2. 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

3. 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

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

5. 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

6. 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

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

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

9. Free-body diagram

10. Chapter 5 Problem 55 A roller coaster at an amusement park has a dip that bottoms out in a vertical circle of radius r. A passenger feels the seat of the car pushing upward on her with a force equal to twice her weight as she goes through the dip. If r = 20.0 m, how fast is the roller coaster traveling at the bottom of the dip?

11. 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

12. Satellites • Accounting for the shape of Earth, projectile motion has to be modified:

13. Satellites • For a circular orbit and the Newton’s Second law • Thus, a speed of a satellite

14. Chapter 5 Problem 33 A satellite is in a circular orbit around an unknown planet. The satellite has a speed of 1.70 × 104 m/s, and the radius of the orbit is 5.25 × 106 m. A second satellite also has a circular orbit around this same planet. The orbit of this second satellite has a radius of 8.60 × 106 m. What is the orbital speed of the second satellite?

15. 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

16. 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

17. 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)

18. 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

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

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

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

22. Chapter 8 Problem 26 A dentist causes the bit of a high-speed drill to accelerate from an angular speed of 1.05 × 104 rad/s to an angular speed of 3.14 × 104 rad/s. In the process, the bit turns through 1.88 × 104 rad. Assuming a constant angular acceleration, how long would it take the bit to reach its maximum speed of 7.85 × 104 rad/s, starting from rest?

23. 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

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

25. Chapter 8 Problem 41 A baseball pitcher throws a baseball horizontally at a linear speed of 42.5 m/s (about 95 mi/h). Before being caught, the baseball travels a horizontal distance of 16.5 m and rotates through an angle of 49.0 rad. The baseball has a radius of 3.67 cm and is rotating about an axis as it travels, much like the earth does. What is the tangential speed of a point on the “equator” of the baseball?

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

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

28. Johannes Kepler (1571-1630) Third Kepler’s law • For a circular orbit and the Newton’s Second law • From the definition of a period

29. Chapter 5 Problem 53 Two satellites, A and B, are in different circular orbits about the earth. The orbital speed of satellite A is three times that of satellite B. Find the ratio (TA/TB) of the periods of the satellites.

30. Questions?