Position: Angular position =! = s r. Displacement: Angular displacement =!" = " 2

Transcription

1 Chapter 11 Rotation Perfectly Rigid Objects fixed shape throughout motion Rotation of rigid bodies about a fixed axis of rotation. In pure rotational motion: every point on the body moves in a circle who s center lies on the rotation axis. Let s set up a convenient coordinate system: Position: Angular position =! = s r [units = rad] Displacement: Angular displacement =!" = " 2 # " 1 Use the Right Hand Rule (RHR) to determine the direction of the angular displacement vector: RHR: fingers curl in direction of rotation Thumb points in direction of vector

2 I. Motion with constant angular velocity A wheel is spinning counterclockwise at a constant rate about a fixed axis. The diagram at right represents a snapshot of the wheel at one instant in time. A. Draw arrows on the diagram to represent the direction of the velocity for each of the points A, B, and C at the instant shown. Is the time taken by points B and C to move through one complete circle greater than, less than, or the same as the time taken by point A? Explain. On the basis of your answer above, determine how the speeds of points A, B and C compare. Explain. B. Mark the position of each of the labeled points at a later time when the wheel has completed one half of a turn. Sketch a velocity vector at each point. For each labeled point, how does the velocity compare to the velocity at the earlier time in part A? Discuss both magnitude and direction. Is there one single linear velocity vector that applies to every point on the wheel at all times? Explain.

3 Rotating object it is pretty annoying to talk about motion in terms of linear velocity (v) since each point on the rotating object has a different velocity. More convenient: find a way to describe the motion of entire rotating object at once, with one single value. Draw position vectors measured from the center of the disk (i.e.: origin at center of disk)

4 C. Suppose the wheel makes one complete revolution in 2 seconds. 1. For each of the following points, find the change in angle (Δθ) of the position vector during one second. (i.e., find the angle between the initial and final position vectors.) Point A Point B Point C 2. Find the rate of change in the angle for any point on the wheel. The rate you calculated above is called the angular speed of the wheel, ω, or equivalently, the magnitude of the angular velocity of the wheel. D. In the space at right sketch the position vectors for point C at the beginning and at the end of a small time interval Δt. 1. Label the change in angle (Δθ) and the distance between the center of the wheel and C (r C ). Sketch the path taken by point C during this time interval. What is the distance that point C travels during Δt? Express your answer in terms of r C and Δθ. 2. Use your answer above and the definition of linear speed to derive an algebraic expression for the linear speed of point C in terms of the angular speed ω of the wheel. What does your equation imply about the relative linear speeds for points farther and farther out on the wheel? Is this consistent with your answers in part A?

5 Angular velocity =! = "# "t Instantaneous angular velocity =! = d" dt rate of change of angular position Use the RHR to denote the direction of the angular velocity vector. Units = rad/s but sometimes you will see units of rev/min, or rpm 1 rev = 2π rad = 360 Relating translational values and angular values: s = r! where s = translational displacement and θ = angular displacement v = r! where v = translational velocity and ω = angular velocity r 2 = 2r 1 ω 1 = 50 rpm ω 2 =? v = speed of a point on the belt = constant On small wheel: v = r 1 ω 1 On large wheel: v = r 2 ω 2 = (2r 1 ) ω 2 v = r 1 ω 1 = (2r 1 ) ω 2 ω 2 = ω 1 /2 = 25 rpm

6 1. Rebecca has gone to a conference in Albany, NY, leaving Brent at home in Dahlonega to look after Barry the dog. a. Which one them has the greater angular speed, ω? b. Which of them has the greater linear speed, v? (Use a diagram showing their positions on the Earth to explain your answers.) 2. Look at an analog clock, like the one on the back wall in the classroom. a. What is the angular speed of the second hand? b. What are the angular speeds of the minute and hour hands? c. Does the size of the clock affect the angular speeds of the hands? Why? d. Does the size of the clock affect the linear speed of the ends of the hands? Why? 3. Look at a block that rests on a rotating platform. a. What affects the 'slipping off' of the block? b. Which way does the block go as it slips off? c. Where is it more likely to slip, and why?

7 If there is a net force acting on the rotating object, then that object will accelerate. Angular acceleration =! = d" = d 2 # dt dt [units = 2 rad/s2 ] Kinematic Equations for rotational motion:!" = # 0 t + 1 $t 2 2! f =! 0 + "t! 2 f =! "#$ Rotating Wheel: The rotational position of a point on the rim of a rotating wheel is given by θ = 4.0t - 2.0t 2 + t 3, where θ is in radians and t is given in seconds. (a) What is the rotational velocity at t = 5 s? (b) What is the rotational velocity at t = 7.0 s? (c) What is the average rotational acceleration for the time interval that begins at t = 5 s and ends at t = 7.0 s? (d) What is the instantaneous rotational acceleration at the beginning of this time interval? (e) What is the instantaneous rotational acceleration at the end of this time interval?

8 A Diver: A diver makes 2.5 revolutions on the way from a 9.1 m high platform to the water. Assuming zero initial vertical velocity, find the diver's average rotational velocity during a dive. A Disk Rotates: Starting from rest, a disk rotates about its central axis with constant rotational acceleration. In 12.0 s, it has rotated 5 rad. (a) What was the rotational acceleration during this time? (b) What was the average rotational velocity? (c) What is the instantaneous rotational velocity of the disk at the end of the 12.0 s? (d) Assuming that the acceleration does not change, through what additional angle will the disk turn during the next 6.0 s?