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

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1 Circular Motion and Gravitation Chapter 7 Angular Measure circle6 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 . circle6 Circular motion is motion in two dimensions circle6 Therefore we can use x-y components to describe it circle6 Sometimes it is more convenient to use polar coordinates Polar Radian Coordinates Measure
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2 Some Geometric Definitions circle6 Definition of the number known as π π C/2R circle6 θ is the measure of the angle subtended by an arc-length S. θ = S/R radian (rad) circle6 We see that when S = R θ = 1 rad circle6 We see that when S = C θ = 2 π rad circle6 Therefore, there are 2 π radians in a complete circle (360 o ) Radian Measure circle6 Angles are calculated in radians θ = S/R circle6 S is the length of arc and R is the radius circle6 Every point on a rigid object undergoes the same angular displacement in any given time interval circle6 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
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3 Angular Small Angle Distance Approximation Example 7-2 circle6 θ = 1.15 o S = 150 m circle6 1.15 o /[2 π rad/360 o ] = 0.0201 rad circle6 R = S/0.0201 = 7473 m = 7.46 km circle6 Or an exact calculation gives: circle6 tan( θ /2) = (L/2)/d circle6 d = (L/2)/tan( θ /2) = 75m/tan(0.0105) = 7.47 km Rotational Motion circle6 Similar development as for linear motion, except use angular quantities circle6 θ = angular distance measured in radians circle6 ω = angular velocity measured in radians/sec. circle6 ω = θ / t circle6 α = angular acceleration in radians/s 2 . circle6 α = ∆ω / t
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4 Linear and Rotational Motion ( With Constant Acceleration) Rotational motion circle6 θ = ω ave t circle6 ω ave = ½ ( ω + ω 0 ) circle6 ω = ϖ 0 + α t circle6 θ = θ 0 + ω 0 t + ½ α t 2 circle6 ω 2 = ω 0 2 + 2 α( α( θ 29 θ 29 θ 29θ Linear motion circle6 x = v ave t circle6 v ave = ½ (v + v 0 ) circle6 v = v 0 + at circle6 x = x 0 + v 0 t + ½ at 2 circle6 v 2 = v 0 2 + 2a( x) The Right-Hand Rule for The Direction of The Angular Velocity vector circle6 Grasp the axis of rotation with your right hand circle6 Wrap your fingers in the direction of rotation circle6 Your thumb points in the direction of θ and ω circle6 You decide on a positive direction. (a) or (b) Tangential and Angular Velocity (Speed) circle6 The velocity (speed) of a particle at a distance r from the axis of rotation is circle6 v = s/ t = r θ / t = r ω circle6 v = r ω circle6 Because it is always tangent to the direction of motion, it is called the tangential velocity (speed)
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5 Frequency and Period of Circular Motion circle6 Let T be the time to rotate through one revolution circle6 T = The period of revolution circle6 i.e. 2 π r vT but v = r ω barb4right
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