Understanding Circular Motion: Concepts and Formulas

1. CHAPTER 4: CIRCULAR MOTION Semester 1 2015 / 2016 DR. MOHD HAFIZUDDIN MAT hafizuddinmat@unimap.edu.my

2. Objectives To define units and angular measure To relate between angular measure and circular arc length To describe and compute angular speed and velocity To explain relationship between angular speed, velocity, and tangential speed.

3. 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.

4. Angular Measure • Arc length of a circle can be related to angular displacement. • Arc length = distance traveled along the circular path. θ = define the arc length. • θ in radians is the ratio of arc length, s and the radius, r.

5. It is most convenient to measure the angle θ in radians:

6. The small-angle approximation is very useful, as it allows the substitution of θ for sin θ when the angle is sufficiently small.

7. Example 1: A spectator standing at the centre of a circular running track observes a runner start a practice race 256 m due east of her own position. The runner runs on the same track to the finish line, which is located due north of the observer’s position. Calculate the distance of the run.

8. Angular Speed and Velocity In analogy to the linear case, we define the average and instantaneous angular speed:

9. Angular Speed and Velocity The direction of the angular velocity is along the axis of rotation, and is given by a right-hand rule.

10. 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

11. 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:

12. Example 2: An amusement park merry go round at its constant operational speed makes one complete rotation in 45 s. Two children are one at 3.0 m from the centre of the ride and the other farther out 6.0 m from centre. Calculate: • Angular speed • The tangential speed of each child

13. 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 towards the center of the circle.

14. Uniform Circular Motion and Centripetal Acceleration The same analysis shows that the centripetal acceleration is given by:

15. 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.

16. Example 3: A space station is in circular orbit about the Earth at an altitude h of 5.0 x 102 km. If the station makes one revolution every 95 min, calculate: • orbital speed • centripetal acceleration

17. Angular Acceleration • The average angular acceleration is the rate at which the angular speed changes: • In analogy to constant linear acceleration

18. 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:

19. Angular Acceleration

20. SUMMARY • To measure the angle θ in radians: • Angular speed • Angular acceleration