Jan 7-11: Mon: Rotational Kinematics (Ch. 8.1 - 8.3) Tue:

1. Jan 7-11: Mon: Rotational Kinematics (Ch. 8.1 - 8.3) Tue: Wed: Torque (Ch. 9.1) Center of Gravity (Ch. 9.3) Thurs: Hookes Law (springs) (Ch. 10.1) Elastic Potential Energy (Ch. 10.3) Fri: Pendulums (Ch. 10.6) Jan 14-18: Review Jan 21: MLK Day (no school) Jan 22-25: Finals Bulls-Eye Lab Practicum

2. Mechanics Wrap-Up Last three weeks of the Semester…

3. 8.1 – 8.3 The angle through which the object rotates is called the angular displacement.

4. Rotational Kinematics Variables 8.3 Linear Motion • x = displacement • v = velocity • a= acceleration • t = time Rotational Motion • θ = displacement (theta) • ω = velocity (omega) • α= acceleration (alpha) • t = time SI Units*: Θ in rad ω in rad/s α in rad/s2 * ω and α might be in rev/s but use dimensional analysis to make sure the equations are “dimensionally balanced”! Counter-clockwise is usually taken to be positive.

5. Rotational Kinematics Equations Linear Kinematics x = x0 + v0 t + ½at2 v = v0 + at x = ½(v0 + v)t v2 = v02 + 2ax v = ½ (v0 + v) Rotational Kinematics θ = θ0+ ω 0t + ½αt2 ω = ω0 + αt θ = ½ (ω0 + ω)t ω2 = ω02 + 2α θ ω = ½ (ω 0 + ω) For CONSTANT linear acceleration For CONSTANT rotational acceleration

6. 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 instead, the blades accelerate and reach a greater angular velocity after the blades have rotated an angular displacement of 44.0 rad. The angular acceleration is constant at 1740 rad/s2. Find the final angular velocity of the blades. (‘Blades’ refers to the TIP of the blade at all times.) Catalogue variables: Using ω2 = ω02 + 2α θ v = 541.98 = 542 rad/s

7. Some Math-y Stuff • If the tip of the blender-blade went around one revolution, how many DEGREES did it travel? • If the tip of the blender-blade went around one revolution, how many RADIANS did it travel? • If the tip of the blender-blade went around 8 revolutions, how many RADIANS did it travel?

8. Math-y Stuff, p. 2 • Recall s = rθ whenθ is in radians • Example: Find the arclength between two points on the circle if the radius is 4.23 x 107 m and the angle is 2 degrees. Convert 2 degrees into radians. Then s = 1.48 x 106 meters

9. Math-y Stuff, p. 3 In many of these problems, use proportionalities. And use dimensional analysis to set up your problems! And finally, think about them! You often ‘know’ how to solve these. For instance, a circle has a radius of 4 meters. An object travels on this circular path, and travels a distance of 11 meters. How many radians did it travel? The distance around the circle one time is 2π4 = 8π= 25.13 m Proportion: 11/25.13 meters = θ/2 π = 2.75 radians (=158 deg.)

10. Conceptual Example A Total Eclipse of the Sun The diameter of the sun is about 400 times greater than that of the moon. By coincidence, the sun is also about 400 times farther from the earth than is the moon. For an observer on the earth, compare the angle subtended by the moon to the angle subtended by the sun and explain why this result leads to a total solar eclipse. Using s = rθ, we can see that the angle is the same in both cases, so the moon is able to completely block the sun!

11. DEFINITION OF AVERAGE ANGULAR VELOCITY

12. Example Gymnast on a High Bar A gymnast on a high bar swings through two revolutions in a time of 1.90 s. Find the average angular velocity of the gymnast.

13. DEFINITION OF AVERAGE ANGULAR ACCELERATION Changing angular velocity means that an angular acceleration is occurring. Example A Jet Revving Its Engines As seen from the front of the engine, the fan blades are rotating with an angular speed 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, assuming it to be constant.

14. Assignment • P. 229 Check Understanding #6 • P. 240 Focus #3-7 • P. 241 Problems #20-25 • Optional challenge #28.