How is rotational motion related to linear motion? HW:

Angular Acceleration How did we determine angular velocity? We simply applied the linear velocity equation in terms of radians! So what is angular acceleration? Symbol: α Units: rad/sec2 Look familiar?

Tying together relationships What expression related angular displacement to displacement and angular velocity to velocity? How is each relationship similar? So, what do you think is the equation that relates angular and linear acceleration?

The Vector Nature of Angular Variables Right-Hand Rule Grasp the axis of rotation with your right hand, so that your fingers circle the axis in the same sense as the rotation. Your extended thumb points along the axis in the direction of the angular velocity vector. Angular acceleration arises when the angular velocity changes, and the acceleration vector also points along the axis of rotation. The acceleration vector has the same direction as the change in the angular velocity.

Example 1. An Accelerating Car An automobile starts from rest and for 20.0 s has a constant linear acceleration of 0.800 m/s2 to the right. During this period, the tires do not slip. The radius of the tires is 0.330 m. At the end of the 20.0-s interval, what is the angle through which each wheel has rotated?

Concepts & Calculations Example 2. Riding a Mountain Bike A rider on a mountain bike is traveling to the left. Each wheel has an angular velocity of +21.7 rad/s, where, as usual, the plus sign indicates that the wheel is rotating in the counterclockwise direction.

To pass another cyclist, the rider pumps harder, and the angular velocity of the wheels increases from +21.7 to +28.5 rad/s in a time of 3.50 s. Determine the angular acceleration of the wheels. After passing the cyclist, the rider begins to coast, and the angular velocity of the wheels decreases from +28.5 to +15.3 rad/s in a time of 10.7 s. Determine the magnitude and direction of the angular acceleration (assumed constant) of the wheels.

Concepts & Calculations Example 3 Concepts & Calculations Example 3.   A Circular Roadway and the Acceleration of Your Car

Suppose you are driving a car in a counterclockwise direction on a circular road whose radius is r = 390 m (see Figure 8.20). You look at the speedometer and it reads a steady 32 m/s (about 72 mi/h). (a) What is the angular speed of the car? (b) Determine the acceleration (magnitude and direction) of the car. (c) To avoid a rear-end collision with a vehicle ahead, you apply the brakes and reduce your angular speed to 4.9 × 10–2 rad/s in a time of 4.0 s. What is the tangential acceleration (magnitude and direction) of the car?

Making things even easier! A lot of the time, rotational questions give you the angular velocity as “rpm” or “rps”. We can convert this since one rotation is 2π radians! What is 5.5 rps in rad/s? “rps” is simply frequency, so this conversion can be written as:

White Board Activity A centrifuge rotor is accelerated from rest to 20,000 rpm in 30 s. What is the average angular acceleration? Through how many revolutions has the centrifuge rotor turned during its acceleration period, assuming constant angular acceleration?

Example 4. 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.

Summary How is linear acceleration, velocity and displacement related to it’s rotational counterpart? How is frequency related to angular velocity? What is “rps” and “rpm” stand for? What do both represent? l=rθ, v = rω, a = rα ω = 2πf Revolutions per minute and revolutions per second. This is frequency (Hz)