Circular Motion

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Explains circular motion as a continuously accelerated motion. Includes frequency, period angular values **More good stuff available at: www.wsautter.com and http://www.youtube.com/results?search_query=wnsautter&aq=f

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Circular Motion

Circular Motion:

Circular Motion Circular motion (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.

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= Angular Displacement ( radians or revolutions )   = Angular velocity ( radians / sec )  = Angular Displacement ( radians / sec ) 2 hz = frequency ( hertz ) sec = Period ( 1 / seconds ) -1 Angular Measurement Units & Symbols

Circular Motion:

Circular Motion The equations 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: V AVERAGE = s/ t = (V 2 + V 1 ) / 2 S i = V 0 t + ½ at 2 V i = V O + at S i = ½ (Vi 2 – V o 2 ) /a Each of these can be converted to a rotational motion equation by substituting the rotational quantity in for the appropriate linear quantity.

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V AVERAGE =  s/  t = (V 2 + V 1 ) / 2  AVERAGE =   /  t = ( 2 +  1 ) / 2 S i = V 0 t + ½ at 2  =  o t + ½ t 2 V i = V O + at  i =  o + t S i = ½ (Vi 2 – V o 2 ) /a  I = ½ ( i 2 -  o 2 ) /  Rotational Equations from Linear Equations

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Additional Rotational Equations & Concepts The linear velocity of 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/s 2 , ft/s 2 , etc  = angular acceleration in radians/s 2 R = the radius of the object in meters, feet, etc.

Frequency & Period:

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 .

Revolutions & Radians:

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

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Summary of Rotational Motion Equations  AVERAGE =  /  t = (  2 +  1 ) / 2  =  o t + ½  t 2  i =  o +  t  i = ½ (  i 2 -  o 2 ) /  s =  R V linear =  R a linear =  R f = 1/ T T = 1 / f 1 revolution = 360 degrees = 2  radians  = 2  f  = 2  / T

Problems in Rotational Motion 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 ? :

Problems in Rotational Motion 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)

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Circular Motion Time for 10 rotations 54 seconds Radius = 3.25 cm What is the frequency ? F = 10 rev / 54 sec = 0.185 Hz What is the period ? T = 1 / f T = 1 / 0.185 = 5.4 seconds What is the angular velocity ?  = 2  f  = 2  (0.185) = 0.37 rad / sec What is the linear distance traveled in one rotation ? S = 2  3.25 = 6.50  cm

Problems in Rotational Motion A CD rotating at 3.5 rad / s makes three revolutions before stopping. (a) What is its angular acceleration (b) How long does it take to stop?:

Problems in Rotational Motion A CD rotating at 3.5 rad / s makes three revolutions before stopping. (a) What is its angular acceleration (b) How long does it take to stop? (a)  o = 3.5 rad/s ,  o = 0 rad/s ,  = 3 rev = 3 x 2  = 6  rad  i = ½ (  i 2 -  o 2 ) / ,  = ½ (  i 2 -  o 2 ) /   = ½ (0 2 – 3.5 2 ) / 6 = -0.325 rad / s 2 (b)  i =  o +  t, t = (  i -  o ) /  t = ( 0– 3.5 ) / -0.325 = 10.8 sec 3.5 rad/s 0 rad/s

Problems in Rotational Motion How many centimeters does the minute hand on a clock move in 25 minutes if it is 40 centimeters long ?:

Problems in Rotational Motion How many centimeters does the minute hand on a clock move in 25 minutes if it is 40 centimeters long ? How many revolutions is 25 minutes? If 60 minutes is one revolution then 25 /60 = 0.417 rev. 0.417 x 2  = 2.62 radians s =  R, s = 2.62 x 40 cm = 105 cm 40 cm ?

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Now it's time for you to try some problems on your own ! The problems are similar to the ones which have been solved so look back and review the appropriate problem if you get stuck !

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What is the linear velocity of a wheel which is 80 cm in diameter if it is turning at 90 rpm ? (A) 0.6 2  m/s (B) 1.2 2  m/s (C) 1.5 2  m/s (D) 2.4 2  m/s A disk is uniformly accelerated from rest to a velocity of 30 rad/s in 8.0 seconds. Through what angle in radians does it turn ? (A) 60 rad (B) 120 rad (C) 240 rad (D) 3600 rad A pulley slows form 150 to 50 rad/s in 6.0 seconds. Through how many radians does it turn ? (A) 300 rad (B) 600 rad (C) 1200 rad (D) 3600 rad What is the angular velocity of a wheel of radius 50 cm which travels at 20 km/h ? (A) 11.1 rad /s (B) 1100 rad /s (C) 0.222 rad /s (D) 1000 rad /s A tire rotating at 2000 rpm requires 50 seconds to stop. What is its angular acceleration ? (A) – 4.2 rad/s 2 (B) - 1.1 rad /s 2 (C) 2.1 rad /s 2 (D) -5250 rad/s 2 Click here for answers

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