Chapter 8. Rotational Kinematics

Transcription

1 Chapter 8 Rotational Kinematics

2 8.1 Rotational Motion and Angular Displacement In the simplest kind of rotation, points on a rigid object move on circular paths around an axis of rotation.

3 8.1 Rotational Motion and Angular Displacement DEFINITION OF ANGULAR DISPLACEMENT When a rigid body rotates about a fixed axis, the angular displacement is the angle swept out by a line passing through any point on the body and intersecting the axis of rotation perpendicularly. By convention, the angular displacement is positive if it is counterclockwise and negative if it is clockwise. θ θ θ o SI Unit of Angular Displacement: radian (rad)

4 8.1 Rotational Motion and Angular Displacement θ (in radians) Arc length Radius s r For a full revolution: 2π r θ r 2π rad 2 π rad 360

5 8.2 Angular Velocity and Angular Acceleration DEFINITION OF AVERAGE ANGULAR VELOCITY Average angular velocity Angular displacement Elapsed time ω θ θo t t o θ t SI Unit of Angular Velocity: radian per second (rad/s) Direction? Clockwise and Counter clockwise

6 8.2 Angular Velocity and Angular Acceleration Example 3 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. ω θ θo t t o θ t θθ 2 rrrrrr 4ππ rrrrrr ωω 4ππ rrrrrr 1.9 ss 6.61 rad/s

7 8.2 Angular Velocity and Angular Acceleration Changing angular velocity means that an angular acceleration is occurring. DEFINITION OF AVERAGE ANGULAR ACCELERATION Average angular acceleration α ω t ω t o o Change in angular velocity Elapsed time ω t SI Unit of Angular acceleration? rad/s 2 Direction? Same as the direction of change in angular velocity

8 8.2 Angular Velocity and Angular Acceleration Example 4 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. α ω ωo t t o ω t rad/s 2

9 8.3 The Equations of Rotational Kinematics Recall the equations of kinematics for constant acceleration. Five kinematic variables: 1. displacement, x 2. acceleration (constant), a 3. final velocity (at time t), v v v + o at ( v)t 1 x 2 vo v vo + 2ax 4. initial velocity, v o 5. elapsed time, t x vo t at 2

10 8.3 The Equations of Rotational Kinematics The equations of rotational kinematics for constant angular acceleration: ANGULAR ACCELERATION ANGULAR VELOCITY ω ω o + αt θ 1 ( ω ω 2 o + )t TIME ANGULAR DISPLACEMENT 2 2 ω ω + 2αθ o θ ω o t + α t

11 8.3 The Equations of Rotational Kinematics

12 8.3 The Equations of Rotational Kinematics Reasoning Strategy Make a free body drawing. Decide which directions are to be called positive (+) and negative (-). (The text uses CCW to be positive.) Write down the values that are given for any of the five kinematic variables. Verify that the information contains values for at least three of the five kinematic variables. Select the appropriate equation. When the motion is divided into segments, remember that the final angular velocity of one segment is the initial velocity for the next. Keep in mind that there may be two possible answers to a kinematics problem.

13 8.3 The Equations of Rotational Kinematics Example 5 Blending with a Blender The blades are whirling with an angular velocity of +375 rad/s when the puree button is pushed in. When the blend button is pushed, 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. 2 2 ω ω + 2αθ o 542 rad/s

14 8.4 Angular Variables and Tangential Variables v T tangential velocity v T tangential speed

15 8.4 Angular Variables and Tangential Variables v T s t rθ θ r t t θ ω t v T rω (ω in rad/s)

16 8.4 Angular Variables and Tangential Variables ( ) ( ) t r t r r t v v a o o To T T ω ω ω ω t ω o ω α ) in rad/s ( 2 α a T rα

17 8.5 Centripetal Acceleration and Tangential Acceleration a ( rω) 2 2 vt c r r 2 rω ( ω in rad/s) a T rα ( α in rad/s 2 )

18 8.5 Centripetal Acceleration and Tangential Acceleration Example 7 A Discus Thrower Starting from rest, the thrower accelerates the discus to a final angular speed of rad/s in a time of s before releasing it. During the acceleration, the discus moves in a circular arc of radius m. Find the magnitude of the total acceleration. ω ω o + αt SSSSSSSSSS ffffff αα a T rα r m and 2 a c rω aa (aa TT 2 + aa cc 2 ) tan 1 ( aa TT aa cc )

19 8.6 Rolling Motion The tangential speed of a point on the outer edge of the tire is equal to the speed of the car over the ground. v rω a rα

20 8.6 Rolling Motion Example 8 An Accelerating Car Starting from rest, the car accelerates for 20.0 s with a constant linear acceleration of m/s 2. The radius of the tires is m. What is the angle through which each wheel has rotated? ωo ω ω 2αθ a rα ω ω o + αt o Solve for θ

21 For Practice FOC Questions: 3, 4, 6, 10, 13 and 15 Problems: 1, 5, 9, 16 and 25