Chapt 7

# Chapt 7 - Circular Motion and Gravitation Chapter 7 Angular...

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1 Circular Motion and Gravitation Chapter 7 Angular Measure c We want to take up the study of Gravitation, but in order to understand the development we need to define some ideas about circular motion . c Circular motion is motion in two dimensions c Therefore we can use x-y components to describe it c Sometimes it is more convenient to use polar coordinates Polar Radian Coordinates Measure

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2 Some Geometric Definitions c Definition of the number known as π π C/2R c θ is the measure of the angle subtended by an arc-length S. θ = S/R radian (rad) c We see that when S = R θ = 1 rad c We see that when S = C θ = 2 π rad c Therefore, there are 2 π radians in a complete circle (360 o ) Radian Measure c Angles are calculated in radians θ = S/R c S is the length of arc and R is the radius c Every point on a rigid object undergoes the same angular displacement in any given time interval c This is why θ is such a natural unit to use in describing circular motion Example 7-1 S = R . θ R = 256m θ = π /2 rad S = 256m . π /2 = 402m
3 Angular Small Angle Distance Approximation Example 7-2 c θ = 1.15 o S = 150 m c 1.15 o /[2 π rad/360 o ] = 0.0201 rad c R = S/0.0201 = 7473 m = 7.46 km c Or an exact calculation gives: c tan( θ /2) = (L/2)/d c d = (L/2)/tan( θ /2) = 75m/tan(0.0105) = 7.47 km Rotational Motion c Similar development as for linear motion, except use angular quantities c θ = angular distance measured in radians c ω = angular velocity measured in radians/sec. c ω = θ / t c α = angular acceleration in radians/s 2 . c α = ∆ω / t

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4 Linear and Rotational Motion ( With Constant Acceleration) Rotational motion c θ = ω ave t c ω ave = ½ ( ω + ω 0 ) c ω = ϖ 0 + α t c θ = θ 0 + ω 0 t + ½ α t 2 c ω 2 = ω 0 2 + 2 α( α( θ29 θ29θ29 Linear motion c x = v ave t c v ave = ½ (v + v 0 ) c v = v 0 + at c x = x 0 + v 0 t + ½ at 2 c v 2 = v 0 2 + 2a( x) The Right-Hand Rule for The Direction of The Angular Velocity vector c Grasp the axis of rotation with your right hand c Wrap your fingers in the direction of rotation c Your thumb points in the direction of θ and ω c You decide on a positive direction. (a) or (b) Tangential and Angular Velocity (Speed) c The velocity (speed) of a particle at a distance r from the axis of rotation is c v = s/ t = r θ / t = r ω c v = r ω c Because it is always tangent to the direction of motion, it is called the tangential velocity (speed)
5 Frequency and Period of Circular Motion c Let T be the time to rotate through one revolution c T = The

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## This note was uploaded on 01/05/2012 for the course PHYS 1401 taught by Professor Jamesr.boyd during the Spring '09 term at Collins.

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Chapt 7 - Circular Motion and Gravitation Chapter 7 Angular...

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