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Angular Motion

Angular Motion. Brad Miller Commonwealth Governor’s School November 6 2008.

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Angular Motion

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  1. Angular Motion Brad Miller Commonwealth Governor’s School November 6 2008

  2. WIND TURBINES such as these can generate significant energy in a way that is environ-mentally friendly and renewable. The concepts of rotational acceleration, angular velocity, angular displacement, rotational inertia, and other topics discussed in this chapter are useful in describing the operation of wind turbines.

  3. Objectives: After completing this module, you should be able to: • Define and apply concepts of angular displacement, velocity, and acceleration. • Draw analogies relating rotational-motion parameters (, , ) to linear (x, v, a) and solve rotational problems. • Write and apply relationships between linear and angular parameters.

  4. A Rotational Displacement,  Consider a disk that rotates from A to B: B Angular displacement q:  Measured in revolutions, degrees, or radians. 1 rev=360 0= 2rad The best measure for rotation of rigid bodies is the radian.

  5. s R R 1 rad = = 57.30 Definition of the Radian One radianis the angle subtended at the center of a circle by an arc length s equal to the radius R of the circle.

  6. R h = 20 m Example 1: A rope is wrapped many times around a drum of radius 50 cm. How many revolutions of the drum are required to raise a bucket to a height of 20 m?

  7. Example 2: A bicycle tire has a radius of 25 cm. If the wheel makes 400 rev, how far will the bike have traveled?

  8. q t w =Angular velocity in rad/s. Angular Velocity Angular velocity, w, is the rate of change in angular displacement. (radians per second.) Angular velocity can also be given as the frequency of revolution, f (rev/s or rpm): w = 2pfAngular frequency f (rev/s).

  9. R h = 10 m Example 3: A rope is wrapped many times around a drum of radius 20 cm. What is the angular velocity of the drum if it lifts the bucket to 10 m in 5 s?

  10. R h = 10 m Example 4: In the previous example, what is the frequency of revolution for the drum? Recall that w =10.0 rad/s.

  11. Angular Acceleration Angular acceleration is the rate of change in angular velocity. (Radians per sec per sec.) The angular acceleration can also be found from the change in frequency, as follows:

  12. R h = 20 m Example 5: The block is lifted from rest until the angular velocity of the drum is 16 rad/s after a time of 4 s. What is the average angular acceleration?

  13. v = w R Angular and Linear Speed From the definition of angular displacement: s = q R Linear vs. angular displacement Linear speed = angular speed x radius

  14. a = aR Angular and Linear Acceleration: From the velocity relationship we have: v = wR Linear vs. angular velocity Linear accel. = angular accel. x radius

  15. R1 A B R2 Examples: Consider flat rotating disk: wo= 0; wf = 20 rad/s t = 4 s R1 = 20 cm R2 = 40 cm What is final linear speed at points A and B? vAf = wAf R1= (20 rad/s)(0.2 m); vAf= 4m/s vAf = wBf R1= (20 rad/s)(0.4 m); vBf= 8m/s

  16. Consider flat rotating disk: R1 A wo= 0; wf = 20 rad/s t = 4 s B R2 R1 = 20 cm R2 = 40 cm Acceleration Example What is the average angular and linear acceleration at B? a = 5.00 rad/s2 a= 2.00 m/s2 a = aR = (5 rad/s2)(0.4 m)

  17. Recall the definition of linear acceleration a from kinematics. becomes Angular acceleration is the time rate of change in angular velocity. Angular vs. Linear Parameters But, a = aR and v = wR, so that we may write:

  18. A Comparison: Linear vs. Angular

  19. R Select Equation: -(62.8 rad/s)2 2(314 rad) 0 - wo2 2q a = = Angular analogy: A disk (R = 50 cm), rotating at 600 rev/min comes to a stop after making 50 rev. What is the acceleration? wo = 600 rpm wf = 0 rpm q = 50rev 50 rev = 314 rad a= -6.29 m/s2

  20. +a R Example 6: A drum is rotating clockwise initially at 100rpm and undergoes a constant counterclockwise acceleration of 3 rad/s2 for 2 s. What is the angular displacement?

  21. Centripetal Acceleration Centripetal means “center seeking” in Latin. Any acceleration that is constantly accelerating an object towards the center of motion is defined as centripetal acceleration and moves the object in a circle. The formula is:

  22. Another Relationship

  23. Forces of Centripetal Acceleration We have learned new ways to look at acceleration; angularly and centripetally. Lets now apply this to Newton’s Second Law:

  24. Example Problem: A 55kg skater is moving 4m/s when she grabs the loose end of a rope tied to a pole. She then moves in a circle radius .8m around the pole. Determine the horizontal force exerted on her arms by the rope.

  25. Summary of Formulas for Rotation

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