An angular/linear speed bicycle example

1 An angular /linear speed bicycle exampleOn October 1, 2003, Leontien Zijlaand-van Moorsel set a new women s hour record byriding a bicycle km in one hour on the velodrome at rode afixed gearbike which was qualitatively like this one:radius 34cmrear sprocket14 teethradius 3cmfront sprocket54 teethrear wheelA fixed gear means that there is no freewheel: the rear sprocket isattached directly to therear wheel, so that if the wheel turns, the rear sprocket (and hencethe front sprocket andpedals) turns. You can t coast on such a bike. These kinds of bikes are standard in trackracing. They also have no brakes, to make it difficult to make sudden speed changes. Thisimprovessafety in the close quarters of track is the question:If Leontien rode at a constant speed , how fast did she pedal?

2 Thatis, how quickly must her pedals (and feet) have been going around?In cycling, this rate isknown as s the idea: if we know how fast the wheels turn, then we ll knowhow fast the rearsprocket turns, then we ll know how fast the chain moves, then we ll know how fast the frontsprocket turns, then we ll know how fast the pedals wheel, sprocket, gear, etc., that turns has both anangular speedand alinear speed :The angular speed is the rate at which the thing turns, described in units like revolutionsper minute, degrees per second, radians per hour, linear speed is the speed at which a a point on the edge of the object travels in itscircular path around the center of the object. The units can be any usual speed units: metersper second, miles per hour, the linear speed of a rotating object,rits radius, and its angular velocity inunits of radians per unit of time, thenv=r.

3 This is an extremely useful formula: it related these three quantities,so that knowing two wecan always find the , the linear speed of a wheel rolling along the ground is also the speed at which thewheel moves along the ground. So if we assume that Leontien moved ata constant speed , thenher wheels were always moving km/hr, (1000m1km)(1hr3600sec)= is the linear speed of her the rear wheel has a radius ofr= , the angular speed of the rear wheel isgiven by =vr= the rear sprocket is attached directly to the rear wheel, it rotates exactly as the rearwheel does: every revolution of the rear wheel is a revolution of thesprocket. Hence, theangular speed of the rear sprocket is rs= that the radius of the rear sprocket is m, we can calculate the linear speed ofthe rear sprocket:vrs= rsrrs= ( )( ) = point in a sprocket-chain system moves at the same linear speed .

4 Hence every pointon the chain has a (linear) speed of m/sec, and the front sprocket has a linear speed ofvfs= now need the radius of the front sprocket in order to find its angular speed . We can usethe fact that the number of teeth on a sprocket must be proportional to itscircumference (so,for instance, if we double the circumference of the sprocket, we double the number of teeth).Thus,54rfs=15rrs= thatrfs=54( )15= this, we calculate the angular speed of the front sprocket: fs=vfsrfs= this into more convenient units, we have fs= Leontien was pedalling about 100 revolutions per minute .