Circular Motion

Circular Motion

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  Sautter 2015
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  Circular Motion • Circularmotion (rotation) can be measured using linear units or angular units. Angular units refer to revolutions, degrees or radians. • The properties of circular motion include displacement, velocity and acceleration. When applied to rotation the values become angular displacement, angular velocity or angular acceleration. Additionally, angular motion can be measured using frequencies and periods or rotation. • The Greek letters theta (), omega () and alpha () are used to represent angular displacement, angular velocity and angular acceleration. 2
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     3
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  Circular Motion • Theequations which describe angular motion are similar to those describing linear motion. Rotational equations can therefore be derived from linear equations by analogy (direct comparison). • Recall the following linear motion equations: • VAVERAGE = s/ t = (V2 + V1) / 2 • Si= V0 t + ½ at2 • Vi = VO + at • Si = ½ (Vi 2 – Vo 2) /a • Each of these can be converted to a rotational motion equation by substituting the rotational quantity in for the appropriate linear quantity. 4
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  VAVERAGE = s/t = (V2 + V1) / 2 AVERAGE =   /  t = (2 + 1) / 2 Si= V0 t + ½ at2  = o t + ½ t2 Vi = VO + at i = o + t Si = ½ (Vi 2 – Vo 2) /a I = ½ (i 2 - o 2) /  5
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  The linear velocityof a body can be related to it angular velocity by the equation: V =  R Where V = linear velocity in m/s , ft/s, etc = angular velocity in radians/s R = the radius of the object in meters, feet, etc. The linear acceleration of a body can be related to it angular acceleration by the equation: a =  R Where a = linear acceleration in m/s2 , ft/s2, etc  = angular acceleration in radians/s2 R = the radius of the object in meters, feet, etc. 6
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  Frequency & Period •Two other important quantities relating to circular motion are frequency and period. • Frequency refers to how often (frequently) an object rotates. If a body rotates 10 complete revolutions in 2 seconds the frequency of rotation is 10/2 or 5 revolutions per second. The unit “hertz” is use to represent cycles or rotations per second. Therefore, 5 revolutions per second is 5 hertz abbreviated as 5 Hz. • Period is the time for one complete cycle or revolution. If an object rotates at 5 revolutions per second (5 Hz) the each revolution takes 1/5 second or 0.20 seconds and the period then is 0.20 seconds. • The symbol for frequency is f. The symbol for period is T. Frequency and period are related to each other by the equation: f = 1/ T and T = 1 / f . 7
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  Revolutions & Radians •Radians are defined as arc length divided by radius length. In a complete circle the circumference is the arc length and is calculated by the equation C = 2  R where R is the radius. Dividing the arc (the circumference) by the radius, we get 2  R / R gives 2  . The number of radians in one complete circle then is 2  . • 1 revolution = 360 degrees = 2  radians • Linear distance can be calculated by multiplying the angular displacement times the radius. • s =  R • Frequency is the number of revolutions per second and since each revolution is 2  radians, radians per second can be calculated by 2  x frequency. Angular velocity is measured in radians per second therefore, angular velocity can be calculated as 2  f. •  = 2  f and since f = 1 / T so  = 2  / T 8
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  AVERAGE = /  t = (2 + 1) / 2  = o t + ½ t2 i = o + t i = ½ (i 2 - o 2) /  s =  R Vlinear =  R alinear =  R f = 1/ T T = 1 / f 1 revolution = 360 degrees = 2  radians  = 2  f  = 2  / T 9
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  Problems in RotationalMotion A wheel 80 cm in diameter turns at 120 rpm (revolution per second) (a) What is its angular velocity (b) What is its linear velocity ? • (a)  = 2  f ,  = 2  2 = 4  radians / sec • (b) V =  R, V = 4  (80) = 320  cm / sec 80 cm 120 rpm Time units in minutes must be converted to seconds (MKS) 120 rpm / 60 = 2 rps rps (revolutions per second) 10
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