Chapter 7 – Angular Motion Things that turn have both a linear velocity and an angular velocity.

tire on a car or bikefilm on a projector buckets on a waterwheelblade on a lawnmower teeth on a gearEarth around the sun propeller on an airplaneseat on a Ferris wheel earth on its axisrope around a pulley fins on a fan or a windmill record on a record player can on a kitchen cabinet lazy susan drum/barrel in a clothes dryer horse on a Merry-Go-Round wind turbines Things that Turn - Examples

Rotational Displacement,  Consider a disk that rotates from A to B: A B  Angular displacement  : can be measured in 1 rev = The best measure for rotation of rigid bodies is the

Definition of the Radian One radian is the angle  subtended at the center of a circle by an arc length s equal to   = = RRRR s or s =  R

Example 1: A rope is wrapped many times around a drum of radius 50 cm. How many revolutions of the drum are required to raise a bucket to a height of 20 m? h = 20 m R

Example 2: A bicycle tire has a radius of 25 cm. If the wheel makes 400 rev, how far will the bike have traveled? Example 2: A bicycle tire has a radius of 25 cm. If the wheel makes 400 rev, how far will the bike have traveled? R = 25cm  = 400 rev s = ? s = R 

Angular Velocity Angular velocity,  is the rate of change in. (radians per second.)  Angular frequency f (rev/s). Angular velocity can also be given as the frequency of revolution, f (rev/s or rpm):  Angular velocity in rad/s.  Angular velocity in rad/s. tttt

Example 3: A rope is wrapped many times around a drum of radius 20 cm. What is the angular velocity of the drum if it lifts the bucket to 10 m in 5 s? R = 20cm s = 10m t = 5s  = ? h = 10 m R

Example 4: In the previous example, what is the frequency of revolution (in rpm) for the drum? Recall that  = 10.0 rad/s. h = 10 m R

Angular Acceleration Angular acceleration is the rate of change in. (Radians per sec per sec.) The angular acceleration can also be found from the change in frequency, as follows:

Example 5: The block is lifted from rest until the angular velocity of the drum is 16 rad/s after a time of 4 s. What is the average angular acceleration?  o = 0  f = 16 rad/s t = 4s  = ? h = 20 m R

Angular and Linear Speed From the definition of angular displacement: s =  R Linear vs. angular displacement Linear speed = angular speed x radius

Angular and Linear Acceleration: From the velocity relationship we have: v =  R Linear vs. angular velocity Linear accel. = angular accel. x radius

Examples: R 1 = 20 cm R 2 = 40 cm R1R1 R2R2 A B   = 0;  f = 20 rad/s t = 4 s What is final linear speed at points A and B? Consider a flat rotating disk where: v Af =

Acceleration Example R 1 = 20 cm R 2 = 40 cm What is the average angular and linear acceleration at B? R1R1 R2R2 A B   = 0  f = 20 rad/s t = 4 s Consider a flat rotating disk:

Angular vs. Linear Parameters Angular acceleration is the time rate of change in angular velocity. Recall the definition of linear acceleration a from kinematics. But, a =  and v = , so that we may write: becomes

Kinematic Equations: Converting Linear to Angular

Angular example: Angular example: A disk (R = 50 cm), rotating at 600 rev/min comes to a stop after making 50 rev. What is the angular acceleration? R  o = 600 rpm  f = 0 rpm  = 50 rev

Example 6: A drum is rotating clockwise initially at 100 rpm and undergoes a constant counterclockwise acceleration of 3 rad/s 2 for 2 s. What is the angular displacement?  o = -100 rpm; t = 2 s  = +3 rad/s 2  = ? R 

Vectors: θ, ω, & α are all vectors. But they are all going in a circular direction, so what straight line vector can represent their direction? We use the “". - Imagine the axis of rotation. - Now "grab" that axis with your fingers going in the same direction of motion. - Your points in the direction of that vector. (Yes, it is to the plane of motion !)

CONCLUSION and SUMMARY: