2WIND 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.
3Objectives: 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.
4Objectives: (Continued) Define moment of inertia and apply it for several regular objects in rotation.Apply the following concepts to rotation:1. Rotational work, energy, and power2. Rotational kinetic energy and momentum3. Conservation of angular momentum
5Rotational Displacement, Consider a disk that rotates from A to B:BAngular displacement q:AMeasured in revolutions, degrees, or radians.1 rev = = 2 radThe best measure for rotation of rigid bodies is the radian.
6Definition of the Radian One radian is the angle subtended at the center of a circle by an arc length s equal to the radius R of the circle.s1 rad = = 57.30R
7Example 1: A rope is wrapped many times around a drum of radius 50 cm 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?h = 20 mRq = 40 radNow, 1 rev = 2p radq = 6.37 rev
8Example 2: A bicycle tire has a radius of 25 cm Example 2: A bicycle tire has a radius of 25 cm. If the wheel makes 400 rev, how far will the bike have traveled?q = 2513 rads = q R = 2513 rad (0.25 m)s = 628 m
9Angular VelocityAngular velocity,w, is the rate of change in angular displacement. (radians per second.)w = Angular velocity in rad/s.qtAngular velocity can also be given as the frequency of revolution, f (rev/s or rpm):w = 2pf Angular frequency f (rev/s).
10Example 3: A rope is wrapped many times around a drum of radius 20 cm 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?h = 10 mRq = 50 radw = =qt50 rad5 sw = 10.0 rad/s
11Example 4: In the previous example, what is the frequency of revolution for the drum? Recall that w = 10.0 rad/s.h = 10 mROr, since 60 s = 1 min:f = 95.5 rpm
12Angular AccelerationAngular 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:
13Example 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?h = 20 mRa = 4.00 rad/s2
14Angular and Linear Speed From the definition of angular displacement:s = q R Linear vs. angular displacementv = w RLinear speed = angular speed x radius
15Angular and Linear Acceleration: From the velocity relationship we have:v = wR Linear vs. angular velocitya = aRLinear accel. = angular accel. x radius
16Examples: R1 A Consider flat rotating disk: B wo = 0; wf = 20 rad/s t = 4 sR1 = 20 cm R2 = 40 cmWhat is final linear speed at points A and B?vAf = wAf R1 = (20 rad/s)(0.2 m); vAf = 4 m/svAf = wBf R1 = (20 rad/s)(0.4 m); vBf = 8 m/s
17Acceleration Example R1 A Consider flat rotating disk: B wo = 0; wf = 20 rad/s t = 4 sConsider flat rotating disk:R1 = 20 cm R2 = 40 cmWhat is the average angular and linear acceleration at B?a = 5.00 rad/s2a = 2.00 m/s2a = aR = (5 rad/s2)(0.4 m)
18Angular vs. Linear Parameters Recall the definition of linear acceleration a from kinematics.But, a = aR and v = wR, so that we may write:becomesAngular acceleration is the time rate of change in angular velocity.
20Linear Example: A car traveling initially at 20 m/s comes to a stop in a distance of 100 m. What was the acceleration?100 mvo = 20 m/svf = 0 m/sSelect Equation:a = =0 - vo22s-(20 m/s)22(100 m)a = m/s2
21Angular analogy: A disk (R = 50 cm), rotating at 600 rev/min comes to a stop after making 50 rev. What is the acceleration?Select Equation:wo = 600 rpmwf = 0 rpmq = 50 revR50 rev = 314 rada = =0 - wo22q-(62.8 rad/s)22(314 rad)a = m/s2
22Problem Solving Strategy: Draw and label sketch of problem.Indicate + direction of rotation.List givens and state what is to be found.Given: ____, _____, _____ (q,wo,wf,a,t)Find: ____, _____Select equation containing one and not the other of the unknown quantities, and solve for the unknown.
23Given: wo = -100 rpm; t = 2 s a = +2 rad/s2 +a Example 6: A drum is rotating clockwise initially at 100 rpm and undergoes a constant counterclockwise acceleration of 3 rad/s2 for 2 s. What is the angular displacement?Given: wo = -100 rpm; t = 2 s a = +2 rad/s2R+aq = radq = rad + 6 radNet displacement is clockwise (-)