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Physic 110 Lecture 18 from Chapter 7 Sections 1 to 3 Rotational Kinematics Homework Assignment 18: Problems: Chapter 7, Problem 2 on page 218 Chapter 7, Problem 4 on page 219 Chapter 7, Problem 6 on page 219 Chapter 7, Problem 10 on page 219 The Radian The radian is a unit of angular measure The radian can be defined as the arc length s along a circle divided by the radius r s r More About Radians Comparing degrees and radians 1 revolution 360 2 rad Converting from degrees to radians [rad] [deg rees] 180 Example 1: A bicycle with 68 cm diameter tires travels 2.0 km. How many revolutions do the wheels make? Example 1: A bicycle with 68 cm diameter tires travels 2.0 km. How many revolutions do the wheels make? r =34 cm = 0.34 m x = s = 2 km = 2000 m s 2000 m 1 rev 5880 rad 936 rev r 0.34 m 2 rad 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: the angle the object rotates through f i where i is the initial angular position f is the final angular position Average Angular Speed The average angular speed, ω, of a rotating rigid object is the ratio of the angular displacement to the time interval f i av tf ti t Angular Speed, cont. The instantaneous angular speed is defined as the limit of the average speed as the time interval approaches zero Units of angular speed are radians/sec rad/s Speed will be positive if θ is increasing (counterclockwise) Speed will be negative if θ is decreasing (clockwise) Example 2 Find the angular speed of the Earth around the sun in radians per second and degrees per day. Example 2 Earth makes 1 rev around sun in 1 year. 1rev 360 deg 1year 0.986o / day t 1year 1rev 365.25days deg 2 rad 1day 1hr 0.986 2x107 rad / s day 360 deg 24hr 3600s Average Angular Acceleration the ratio of the change in the angular speed to the time it takes f i av tf ti t Units of angular acceleration: rad/s2 Analogies Between Linear and Rotational Motion Uniform motion: ω = constant Linear: Rotational: x f xi v t f o t where θi and θf are the angular displacements and ω is the angular velocity Analogies Between Linear and Rotational Motion Uniform accelerated motion: α= constant Linear: Rotational: v f vo a t f o t x f xo vo v f t f o o f t 1 1 2 2 1 2 1 2 x f xo vot a t f o o t t 2 2 v 2 vo 2a( x f xo ) f 2 2 o2 2 ( f o ) f Example 3: A tire on a balancing machine starts from rest and turns through 4.7 revolutions in 1.2 s before reaching its final speed. Assuming the acceleration was constant, find a) the angular velocity at the end of this time b) the angular acceleration Example 3: θo = 0.0 θf = 4.7 rev = 29.5 rad t = 1.2 s ωo = 0.0 rad/s ωf = ? α=? For uniform angular acceleration: f o o f t 1 2 2 f o f o t 2 29.5rad 0 f 0 49.2rad / s 1.2s Example 3: θo = 0.0 θf = 4.7 rev = 29.5 rad t = 1.2 s ωo = 0.0 rad/s ωf =49.2 rad/s α=? For uniform angular acceleration: f o t f o t 49.2rad / s 0 41rad / s 2 1.2s Relationship Between Angular and Linear Quantities Consider a ball moving arc length s r along an arc. v at velocity along arc an v r r s acceleration along arc ω θ at r α acceleration normal to arc 2 v an r 2 r Example 4: A 7.60 m diameter helicopter rotor rotates at a constant speed of 450 rev/min. What is the speed of the tip? What is the acceleration of the tip? r = 7.6/2 = 3.8 m ω = 450 rev/min = 47.1 rad/s v r 47.1rad / s 3.8m 179 m / s Example 4: A 7.60 m diameter helicopter rotor rotates at a constant speed of 450 rev/min. What is the speed of the tip? What is the acceleration of the tip? (α = 0 rad/s2) at r an r 2 0rad / s 2 3.8m 47.1rad / s (3.8m) 2 0m / s 2 680m / s 2

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angular velocity, Rotational Motion, angular position, Example 2, time interval, Example 1, angular acceleration, Chapter 7, angular speed, Angular Displacement

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posted: | 7/11/2011 |

language: | English |

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