Rotational Kinematics Rotational Motion and Angular Displacement Chapter 8 Lesson 2

Example 4. A Jet Revving Its Engines A jet awaiting clearance for takeoff is momentarily stopped on the runway. As seen from the front of one engine, the fan blades are rotating with an angular velocity of –110 rad/s, where the negative sign indicates a clockwise rotation . As the plane takes off, the angular velocity of the blades reaches –330 rad/s in a time of 14 s. Find the average angular acceleration.                                                                                                                                 

In example 4, assume that the orientation of the rotating object is given by q 0 = 0 rad at time t0 = 0 s. Then, the angular displacement becomes Dq  = q  – q 0 = q , and the time interval becomes Dt = t – t0 = t.

The Equations of Kinematics for Rational and Linear Motion Rotational Motion (a = constant)  Linear Motion (a = constant) 

q Displacement x w0 Initial velocity v0 w Final velocity v a Symbols Used in Rotational and Linear Kinematics Rotational Motion  Quantity  LinearMotion   q   Displacement x w0  Initial velocity  v0  w Final velocity  v a  Acceleration  a t Time 

Example 5. Blending with a Blender The blades of an electric blender are whirling with an angular velocity of +375 rad/s while the “puree” button is pushed in. When the “blend” button is pressed, the blades accelerate and reach a greater angular velocity after the blades have rotated through an angular displacement of +44.0 rad (seven revolutions). The angular acceleration has a constant value of +1740 rad/s2. Find the final angular velocity of the blades.

 q    a   w  w0   t  +44.0  rad  +1740  rad/s2  ?  +375  rad/s                                                                                                                                                                                                 

Angular Variables and Tangential Variables For every individual skater, the vector is drawn tangent to the appropriate circle and, therefore, is called the tangential velocity vT. The magnitude of the tangential velocity is referred to as the tangential speed.

If time is measured relative to t0 = 0 s, the definition of linear acceleration is given by Equation 2.4 as aT = (vT – vT0)/t, where vT and vT0 are the final and initial tangential speeds, respectively.

Example 6. A Helicopter Blade A helicopter blade has an angular speed of w = 6.50 rev/s and an angular acceleration of a = 1.30 rev/s2. For points 1 and 2 on the blade, find the magnitudes of (a) the tangential speeds and (b) the tangential accelerations.

(a)                                                                                                                                                                 (b)

Centripetal Acceleration and Tangential Acceleration

The centripetal acceleration can be expressed in terms of the angular speed w by using vT = rw While the tangential speed is changing, the motion is called nonuniform circular motion. Since the direction and the magnitude of the tangential velocity are both changing, the airplane experiences two acceleration components simultaneously. aT aC

Check Your Understanding 3 The blade of a lawn mower is rotating at an angular speed of 17 rev/s. The tangential speed of the outer edge of the blade is 32 m/s. What is the radius of the blade? 0.30 m

Example 7. A Discus Thrower Discus throwers often warm up by standing with both feet flat on the ground and throwing the discus with a twisting motion of their bodies. A top view of such a warm-up throw. Starting from rest, the thrower accelerates the discus to a final angular velocity of +15.0 rad/s in a time of 0.270 s before releasing it. During the acceleration, the discus moves on a circular arc of radius 0.810 m.

Find (a) the magnitude a of the total acceleration of the discus just before it is released and (b) the angle f that the total acceleration makes with the radius at this moment. (a)                                                                                                                                                                                 

(b)

Check Your Understanding 4 A rotating object starts from rest and has a constant angular acceleration. Three seconds later the centripetal acceleration of a point on the object has a magnitude of 2.0 m/s2. What is the magnitude of the centripetal acceleration of this point six seconds after the motion begins?

(=0) after six second,

at six second 8.0 m/s2