Ying Yi PhD Chapter 8 Rotational Kinematics 1 PHYS HCC

Outline 2 Angular Velocity and Angular Acceleration Rotational Kinematics Equations Relation between Angular and Linear Quantities Rolling Motion PHYS HCC

3 The Radian The radian is a unit of angular measure The radian can be defined as the arc length along a circle divided by the radius r

More About Radians Comparing degrees and radians Converting from degrees to radians 4 PHYS HCC

5 Angular Displacement Axis of rotation is the center of the disk Need a fixed reference line During time t, the reference line moves through angle θ

Angular Displacement, cont. The angular displacement is defined as the angle the object rotates through during some time interval The unit of angular displacement is the radian Each point on the object undergoes the same angular displacement 6 PHYS HCC

Example 8.1: Angular displacement PHYS HCC 7 Synchronous or “stationary "communications satellites are put into an orbit whose radius is r=4.23×10 7 m. The orbit is in the plane of the equator, and two adjacent satellites have an angular separation of Ɵ =2.00º, as Figure 8.4 illustrates. Find the arc length s that separates the satellites.

PHYS HCC 8 Average Angular Velocity The average angular velocity, ω, of a rotating rigid object is the ratio of the angular displacement to the time interval

Example 8.3: Angular velocity PHYS HCC 9 A gymnast on a high bar swings through two revolutions in a time of 1.90 s, as Figure 8.6 suggests. Find the average angular velocity (in rad/s) of the gymnast.

Instantaneous Angular Velocity The instantaneous angular speed is defined as the limit of the average velocity as the time interval approaches zero Units of angular speed are radians/sec rad/s Velocity will be positive if θ is increasing (counterclockwise) Velocity will be negative if θ is decreasing (clockwise) 10 PHYS HCC

Average Angular Acceleration The average angular acceleration of an object is defined as the ratio of the change in the angular speed to the time it takes for the object to undergo the change: 11 PHYS HCC

Angular Acceleration, cont Units of angular acceleration are rad/s² Positive angular accelerations are in the counterclockwise direction and negative accelerations are in the clockwise direction When a rigid object rotates about a fixed axis, every portion of the object has the same angular speed and the same angular acceleration 12 PHYS HCC

Angular Acceleration, final The sign of the acceleration does not have to be the same as the sign of the angular velocity The instantaneous angular acceleration is defined as the limit of the average acceleration as the time interval approaches zero 13 PHYS HCC

14 Linear and Rotational Motion

Example 8.5: Rotational Kinematics PHYS HCC 15 The blades of an electric blender are whirling with an angular velocity of +375 rad/s while the “puree” button is pushed in, as Figure 8.8 shows. 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 rad. The angular acceleration has a constant value of rad/s 2. Find the final angular velocity of the blades.

Group Problem: A rotating Wheel PHYS HCC 16 A wheel rotates with a constant angular acceleration of 3.50 rad/s 2. If the angular speed of the wheel is 2.00 rad/s at t=0, (a) through what angle does the wheel rotate between t=0 and t=2.00s? Give your answer in radians and in revolutions. (b) What is the angular speed of the wheel at t=2.00 s? (c) What angular displacement (in revolutions) results while the angular speed found in part (b) doubles?

Relationship Between Angular and Linear Quantities Displacements Speeds Accelerations Every point on the rotating object has the same angular motion Every point on the rotating object does not have the same linear motion 17 PHYS HCC

Example 8.6: Helicopter Blade PHYS HCC 18

Group Problem: Track Length of a compact disc PHYS HCC 19 In a compact disc player, as the read head moves out from the center of the disc, the angular speed of the disc changes so that the linear speed at the position of the head remains at a constant value of about 1.3 m/s (a) find the angular speed of the compact disc when the read head is at r=2.0 cm and again at r=5.6 cm. (b) An old-fashioned record player rotates at a constant angular speed, so the linear speed of the record groove moving under the detector changes. Find the linear speed of a 45.0 rpm record at points 2.0 and 5.6 cm from the center. (c) In both the CD and phonograph record, information is recorded in a continuous spiral track, 1.3 m/s. Calculate the total length of the track for a CD designed to play for 1.0 h.

Centripetal Acceleration and Tangential Acceleration PHYS HCC 20 Uniform Circular MotionNonuniform Circular Motion

Centripetal Acceleration (review) 21 PHYS HCC

22 Total Acceleration The tangential component of the acceleration is due to changing speed The centripetal component of the acceleration is due to changing direction Total acceleration can be found from these components

PHYS HCC 23 Vector Nature of Angular Quantities Direction can be more completely defined by using the right hand rule Grasp the axis of rotation with your right hand Wrap your fingers in the direction of rotation Your thumb points in the direction of ω

PHYS HCC 24 Velocity Directions, Example In a, the disk rotates clockwise, the velocity is into the page In b, the disk rotates counterclockwise, the velocity is out of the page

Acceleration Directions If the angular acceleration and the angular velocity are in the same direction, the angular speed will increase with time If the angular acceleration and the angular velocity are in opposite directions, the angular speed will decrease with time 25 PHYS HCC

Example 8.7: A Discus Thrower PHYS HCC 26 Discus throwers often warm up by throwing the discus with a twisting motion of their bodies. Figure 8.13a illustrates a top view of such a war-up throw. Starting from rest, the thrower accelerates the discus to a final angular velocity of rad/s in a time of s before releasing it. During the acceleration, the discus moves on a circular arc of radius m. Find the magnitude a of the total acceleration of the discus just before the discus is released.

Group Problem: At the Racetrack PHYS HCC 27 A race car accelerates uniformly from a speed of 40.0 m/s to a speed of 60.0 m/s in 5.00 s while traveling counterclockwise around a circular track of radius 4.00×10 2 m. When the car reaches a speed of 50.0 m/s, find (a) the magnitude of the car’s centripetal acceleration, (b) the angular speed, (c) the magnitude of the tangential acceleration, and (d) the magnitude of the total acceleration.

Rolling Motion PHYS HCC 28

Example 8.8 An accelerating car PHYS HCC 29 An automobile, starting from rest, has a linear acceleration to the right whose magnitude is m/s 2. During the next 20.0 s, the tires roll and do not slip. The radius of each wheel is m. At the end of this time, what is the angle through which each wheel has rotated?

Homework PHYS HCC 30 7,9,13,21,25,29,40,47,48,51,53