Chapter 8 Rotational Kinematics
In the simplest kind of rotation, points on a rigid object move on circular paths around an axis of rotation.
Example Hans Brinker is on skates and there is no friction. He has two identical snowballs, and wants to get from A to B just by throwing the balls. Is it better to throw them together, or one after the other? Assume that the relative velocity after release is the same whether he throws one or two snowballs.
1) Angular Displacement Angle swept by a line intersecting and perpendicular to the axis of rotation Angular position θ = s r Change in angular position Δθ = θ θ 0 CCW ==> +ve (measured in radians)
i>clicker A meter stick is rotated about the end labeled 0.00 cm, so that the other end of the stick moves through an arc length of 8.0 cm. Through what arc length s does the 25-cm mark on the stick move? a) 8.0 cm b) 2.0 cm c) 32.0 cm
i>clicker Over the course of a day (twenty-four hours), what is the angular displacement of the seconds hand of a wrist watch in radians? A. -2π rad B. -2π 1440 rad C. 2π 1440 rad D. -2π 8640 rad E. 2π 8640 rad
2) Angular Velocity Average angular velocity ω = Δθ Δt rad/s or s -1 Instantaneous angular velocity ω = lim Δt 0 Δθ Δt
Period, T time for one revolution ω = 2π T Tangential speed v T = Δs Δt = rδθ Δt v T = rω
Example The earth spins on its axis once a day and orbits the sun once a year (365.25 days). Determine the average angular velocity (in rad/s) of the earth as it spins on its axis and orbits the sky. In each case, take the positive direction for the angular displacement to be the direction of the earth s motion.
i>clicker You are in a tall building located near the equator. As you ride an elevator from the ground floor to the top floor, your tangential speed due to the earth s rotation. a) increases b) decreases c) increases when the speed of the elevator increases and decreases when the speed of the elevator decreases d) does not change
Example Two people start at the same place and walk around a circular lake in opposite directions. One walks with an angular speed of 1.7x10-3 rad/s, while the other has an angular speed of 3.4x10-3 rad/s. How long will it be before they meet?
3) Angular Acceleration Average angular acceleration α = Δω Δt rad/s 2 or s -2 Instantaneous angular velocity α = lim Δt 0 Δω Δt same sign as ang velocity ==> speeding up opp sign ==> slowing down
Tangential acceleration a T = Δv T Δt = Δ(rω ) Δt = r Δω Δt a T = rα
Tangential and angular quantities: multiply by r s = rθ v T = rω a T = rα
i>clicker An electric clock is hanging on the wall in the living room. The battery is removed and the second hand comes to a halt over a brief period of time. Which one of the following statements correctly describes the angular velocity ω and the angular acceleration α of the second hand as it slows down? a) ω and α are both negative. b) ω is positive and α is negative. c) ω is negative and α is positive. d) ω and α are both positive.
Example A CD player has a playing time of 74 minutes. When the music starts, the CD is rotating at an angular speed of 480 revolutions per minute (rpm). At the end of the music, the CD is rotating at 210 rpm. Find the magnitude of the average angular acceleration of the CD player. Express your answer in rad/s 2.
4) Equations of rotational kinematics (for constant angular acceleration) θ = ω 0 t + 1 2 αt 2 ω = ω 0 + αt ω 2 = ω 0 2 + 2αθ θ = ωt 1 2 αt 2 θ = 1 2 (ω 0 + ω)t
Reasoning Strategy 1. Make a drawing. 2. Decide which directions are to be called positive (+) and negative (-). (The text uses CCW to be positive.) 3. Write down the values that are given for any of the five kinematic variables. 4. Verify that the information contains values for at least three of the five kinematic variables. Select the appropriate equation. 5. When the motion is divided into segments, remember that the final angular velocity of one segment is the initial velocity for the next. 6. Keep in mind that there may be two possible answers to a kinematics problem.
Example The shaft of a pump starts from rest and has an angular acceleration of 3.00 rad/s 2 for 18.0 s. At the end of this interval what is the shaft s angular speed and the angle through which the shaft has turned? θ = ω t + 1 αt 2 0 2 ω = ω + αt 0 ω 2 = ω 2 + 2αθ 0 θ = 1 (ω + ω)t 2 0
i>clicker A wheel which is initially at rest starts to turn with a constant angular acceleration. After 4 seconds it has made 4 complete revolutions. How many revolutions has it made after 8 seconds? A. 8 B. 12 C. 16 θ = ω 0 t + 1 2 αt 2 ω = ω 0 + αt ω 2 = ω 0 2 + 2αθ θ = 1 2 (ω 0 + ω)t
Example A spinning wheel on a fireworks display is initially rotating in a counterclockwise direction. The wheel has an angular acceleration of -4.00 rad/s 2. Because of this acceleration, the angular velocity changes from its initial value to a final value of -25.0 rad/s. θ = ω 0 t + 1 2 αt 2 ω = ω 0 + αt ω 2 = ω 2 0 + 2αθ θ = 1 2 (ω 0 + ω)t θ = ωt 1 2 αt 2 The net angular rotation of the wheel is zero. Find the time for the change in the angular velocity to occur.