Chapter 8 Rotational Kinematics

The axis of rotation is the line around which an object rotates.

The angle through which a rigid body rotates about a fixed axis is called the angular displacement.

Angular displacement  can be expressed in degrees, or revolutions, or radians.

The SI unit of angular displacement is the radian (rad).

The formula is  –  0.  (in radians) = arc length/radius

By convention, a counter-clockwise rotation is positive, and a clockwise rotation is negative.

360° = 2  rad So 1 rad = 57.3°.

Example 1. Synchronous satellites orbit at a radius of 4.23 x 10 7 m. Two adjacent satellites have an angular separation of  = 2.00°. Find the arc length that separates the satellites.

Angular velocity is the angular displacement divided by elapsed time.  =  /  t

The unit is radians per second. rad/s

Example 3. A gymnast on a high bar swings through two revolutions in time of 1.90 s. Find the average angular velocity (in rad/s) of the gymnast.

Instantaneous angular velocity is the angular velocity at a given instant.

Angular acceleration  is the rate of change of angular velocity.  =  /  t

Example 4. A jet engine’s turbine fan blades are rotating with an angular velocity of -110 rad/s. As the plane takes off, the angular velocity of the blades reaches -330 rad/s in a time of 14 s. Find the angular acceleration.

The equations for rotational dynamics are similar to those for linear motion.  =  0 +  t

 =  0 t + ½  t 2  2 =   

Example 5. The blades of an electric blender are turning with an angular velocity of +375 rad/s. 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 rad. The angular acceleration has a constant value of rad/s 2. Find the final angular velocity of the blades.

The tangential velocity v T is the speed in m/s around the arc. The magnitude is called the tangential speed. v T = r 

Example 6. A helicopter blade has an angular speed of  = 6.50 rev/s and an angular acceleration of  = 1.30 rev/s 2. For a point 3 m from the center and a point 6.7 m from the center, find (a) the tangential speeds and (b) the tangential accelerations.

The centripetal acceleration formula is a c = v T 2 /r. This can be expressed in terms of angular speed since v T = r .

a c = v T 2 /r becomes a c = (r  ) 2 /r a c = r  2 (  is rad/s)

When the tangential speed is changing, the motion is called nonuniform circular motion. When there is tangential acceleration and centripetal acceleration, the total accelration can be found from the Pythagorean theorem.

Example 7. A discus thrower warming up accelerates the discus to a final angular velocity of rad/s in a time of s before releasing it. During the acceleration, the discus moves on a circular arc of radius m. Find (a) the magnitude a of the total acceleration of the discus just before it is released and (b) the angle that the acceleration makes with the radius at this moment.

When objects roll there is a relationship between the angular speed of the object and the linear speed at which the object moves forward.

Linear speed is equal to tangential speed. v = r  It follows that linear acceleration is equal to tangential acceleration. a = r 

Example 8. A car starts from rest and for 20 s has a constant linear acceleration of 0.8 m/s 2 to the right. The radius of the tires is 0.33 m. At the end of the 20 s interval what is the angle through which the wheel has rotated?

The direction of the angular velocity vector is along the axis of rotation. The direction along that axis is found using the Right-Hand Rule.

Right-Hand Rule. When the fingers of your right hand encircle the axis of rotation, and your fingers point in the direction of the rotation, your extended thumb points in the direction of the angular velocity vector.

The direction of the angular acceleration vector is found the same way. The direction is determined by the change in angular velocity.

If the angular velocity is increasing, the angular acceleration vector points in the same direction as the angular velocity.

If the angular velocity is decreasing, the angular acceleration vector points in the opposite direction as the angular velocity.

Example 9. A rider on a bike is traveling to the left. Each wheel has an angular velocity of rad/s. (a) The angular velocity of the wheels increases from to rad/s in a time of 3.50 s. (b) The rider then begins to coast, and the angular velocity of the wheels decreases from to rad/s in a time of 10.7 s. In each instance, determine the magnitude and direction of the angular acceleration of the wheels.

Example 10. A car is moving counterclockwise on a circular road of radius r = 390 m. The speedometer reads 32 m/s. (a) What is the angular speed of the car? (b) Find the acceleration of the car. (c) The angular speed is reduced to 4.9 x rad/s in a time of 4.0 s. What is the tangential acceleration of the car?