Measuring Rotational Motion

1. Measuring Rotational Motion Chapter 7 Section 1

2. What is Rotational Motion? • Rotational Motion – The motion of a body that spins about an axis. • Examples: • Ferris Wheel • Bicycle Wheel • Merry-go-round • Etc…

3. Circular Motion • A point that rotates about a single axis undergoes circular motion around that axis. • Regardless of the shape of the object, any single point on the object travels in a circle around the axis of rotation.

4. Linear motion here? Think again • Its difficult to describe the motion of a point in a circular path through linear quantities, because the direction of the motion is not in a straight line path. • Therefore it is describe in terms of angle through which the point on an object moves.

5. Rotational Motion • When rotational motion is described using angles, all points on a ridged rotating object, except the points on the axis move through the same angle during any time interval.

6. r s θ Reference Line • r = radius • θ = angle • s = arc length

7. Radians • Radian – An angle whose arc length is equal to its radius, which is approximately equal to 57.3º • The radian is a pure number with no dimensions. • In almost all of the equations in chapter 7 and chapter 8 will use radians instead of degrees.

8. Radian Chart

9. Radian Equations • Any angle θ, measured in radians, is defined as the following: • To convert degrees to radians, or vise versa, use the following equation:

10. Angular Displacement • Angular Displacement – The angle through which a point, line, or body is rotated in a specific direction and about a specified axis. • In simple terms, angular displacement describes how much an object has rotated. • It is not a distance!

11. Angular Displacement Equation • Δθ = angular displacement (in radians) • Units for angular displacement: rad • Δs = change in arc length • r = distance from axis • When Δθ rotates: • Clockwise – Negative • Counter-clockwise – Positive

12. Recall • Recall that Δ is a capital Greek letter, “delta” • It means, “Change in” • It is always the final minus the initial • Example: • Δθ = θf-θi

13. Example Problem • Earth has an equatorial radius of approximately 6380km and rotates 360º in 24 hours. • What is the angular displacement (in degrees) of a person standing on the equator for 1.0 hour? • Convert this angular displacement to radians • What is the arc length traveled by this person?

14. Example Problem Answer • 15º • 0.26 rad • Approximately 1700km

15. Example Problem • Earth has an average distance from the sun of approximately 1.5x10^8 km. For its orbital motion around the sun, find the following: • Average daily angular displacement • Average daily distance traveled

16. Example Problem Answer • 1.72x10^-2 rad • 2.58x10^6 km

17. Angular Speed • Angular Speed – The rate at which a body rotates about an axis, usually expressed in radians per second. (rad/s) • Recall that linear speed describes the distance traveled in a specific time, here it is the angular displacement in a specific time. • It describes how quickly the rotation occurs.

18. Angular Speed Equation • ω = Average angular speed • Lower case Greek letter called, “omega” • Units: radians per second (rad/s) • Δθ = Angular displacement • Δt = Time interval

19. Revolutions • Occasionally angular speeds are given in revolutions per unit time. • 1 rev = 2πrad • Examples: • DVD’s • Records • Engines • 2000 rpm (revolutions per minute)

20. Example Problem • An Indy car can complete 120 laps in 1.5 hours on a circular track. Calculate the average angular speed of the Indy car.

21. Example Problem Answer • 0.14 rad/s

22. Angular Acceleration • Angular Acceleration – The time of change of angular speed, expressed in radians per second per second.

23. Angular Acceleration Equation • α = average angular acceleration • Lower case Greek letter, “alpha” • Units: rad/s² • Δω = Change in angular speed • t = time interval

24. Example Problem • A yo-yo at rest is sent spinning at an angular speed of 12 rev/s in 0.25 seconds. What is the angular acceleration of the yo-yo?

25. Example Problem Answer • 48 rev/s² • Converted to radians • 301.6 rad/s²

26. All Points Have Same Speed and Acceleration • If a point on the rim of a bicycle wheel had a greater angular speed than a point closer to the axel, the shape of the wheel would be changing. • Which can not happen in normal everyday riding. • All points on a rotating object have the same angular speed & acceleration.

27. Comparing Linear and Angular QuantitiesAngular Substitutes for Linear Quantities

28. Kinematic Equations Can be Used for Circular Motion • The same equations used for linear motion with constant acceleration, can be used for circular motion with constant angular acceleration. • Change the variables, by using the table in the previous slide and you have the new kinematic equations.

29. Kinematic Equations for Constant Angular AccelerationAngularLinear

30. Example Problem • A barrel is given a downhill rolling start of 1.5rad/s at the top of a hill. Assume a constant angular acceleration of 2.9rad/s² • If it takes 11.5s to get to the bottom of the hill, what is the final angular speed of the barrel? • What angular displacement does the barrel experience during the 11.5s ride?

31. Example Problem Answers • 35 rad/s • 210 rad