STATEMENT OF THE PROBLEM The Smoky Hills Wind Farm near Salina, Kansas employs 1
ID: 1781715 • Letter: S
Question
STATEMENT OF THE PROBLEM The Smoky Hills Wind Farm near Salina, Kansas employs 140,000 pound Vestas V80 1.8 megawatt wind turbines. The three-blade turbines have a diameter of 80 mand operate at 15.5 to 16.8 rpm (revolutions per minute.) a) What is the linear speed of the blade tip at maximum rotational speed? b) What is the radial (centripetal) acceleration at the tip of the blade at the maximum speed? c) If the blade slows down from maximum speed to rest in 30 seconds, through how many revolutions does it turn? STRATEG!Y We use the relation: linear quantity = (radius) times (corresponding angular quantity) arc length = (radius) times (angle subtended) linear speed along the arc = (radius) times (angular speed) tangential acceleration = (radius) times (angular acceleration) The above relations are valid if the angles are measured in radians. The tangential acceleration ae is non-zero if the angular speed is changing. Whenever the angular speed is non-zero, there is always a radial (centripetal) acceleration, responsible for changing the direction of the tangential velocity. IMPLEMENTATION Questions 1 & 2 Question 3 The angular speed is given in revolutions per minute. Since there are 2 radians in a revolution and 60 seconds in a minute, we multiply rpms by 21t/60 to get the angular speed in radians per second. To obtain the tangential speed, we use v-T. The radial (centripetal) acceleration is then a,--. in 140,000 lbs d= 80 m The kinematics equations for angular quantities mimic kinematics equations for linear motion. For constant angular acceleration or, the relation between , a), and is:Explanation / Answer
maximum rotational speed is w =16.8 rpm = 16.8*(2*3.142/60) =1.76 rad/s
a) linear speed is v = r*w = (diameter/2)*1.76 = (80/2)*1.76 =70.4 m/sec
b) radial accelaration is a_rad = v^2/r = 70.4^2/(80/2) = 123.9 m/s^2
c) using
w^2-wi^2 = 2*alpha*theta
0^2 - 1.76^2 = 2*alpha*theta
angular accelaration is alpha = (-w/t) = -1.76/30 = -0.0586 rad/s^2
then
-1.76^2 = -2*0.0586*theta
theta = 26.43 rad
so no.of revolutions are 26.43/(2*3.142) = 4.2 rev
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