The document discusses the physics behind how bicycles work. It explains that pedaling provides torque that drives the rear wheel forward via the chain and gears. Having multiple gears allows riders to maintain an efficient cadence across different terrains and slopes. The stability of bicycles comes primarily from their trail geometry, where the front wheel contact point trails behind the steering axis. This causes any lean to self-correct, keeping the bicycle upright.

1. The Physics Of
Bicycles
Daniel Eley

2.  First chain driven bicycle introduced in 1885
 Structure hasn’t changed much
 How do they work?
John Starley’s “Rover” of 1885
Lance Armstrong’s 2004
Tour de France bike

3. Topics of Discussion
 Why does the bike move forward
when I pedal?
 What are gear ratios?
 What determines a bike’s stability?

4. A Moving Bicycle
R = radius of the tires w = angular velocity of the tires
r1 = radius of the sprocket r2 = radius of the chainring
N1 = normal force N2 = normal force
M = total mass of bicycle and rider p = length of pedal arm
F = force applied vertically at a right angle to the pedal arm
F1 = reactive push backwards from ground
F2 = drive force forwards due to torque from pedaling

5. We know…
Fnet x = Ma (where a is the tangential acceleration)
a = aR (where a is the angular acceleration of the tires)
F2 – F1 = Fnet x
Therefore…
F2 – F1 = MRa (1)

6. tnet = Ia
(where t is the torque and I is the moment of inertia of the back tire)
Fp(r1/r2) – F2R = Ia (2)
If we assume that the moment of inertia of the front wheel is
roughly the same as the back wheel, then we can say…
F1R = Ia (3)
We combine equations 1,2, and 3 eliminating F1 and F2
Fp(r1/r2) = a(2I + MR2)

7. Using this equation and a=Ra, the bicycle’s linear acceleration is:
a = RFp(r1/r2)
(2I+MR2) (4)
the smaller the gear ratio (r2/r1), the larger the acceleration
So why don’t all bikes have a small r2/r1 ratio?

8.  angular velocity of the sprocket (r1) be w1 (also equal to w)
 angular velocity of the chainring (r2) be w2
 the velocity of the foot applied to the pedal is…
vapp = pw2 or w2 = vapp /p
The velocity of the bike is…
vb = Rw = Rw1 = w2(r2/r1)R
Substituting vapp /p for w2 and rearranging gives us…
vb / vapp = r2R/r1p

9. vb / vapp = (r2/r1)R/p (6)
 to maintain a constant velocity (vb), and a minimum vapp, you
would want a large gear ratio (r2/r1).
 But as we saw in eq. 4, for a higher acceleration, you want a
small gear ratio (r2/r1).
 How do you give riders both?
 Multiple sprockets or gears

10. Bicycle Gearing
 The gearing on a bicycle is the
selection of appropriate gear ratios
for optimum efficiency or comfort.
 The gear ratio is the ratio of the
number of teeth on the chainring of
the crankset to the teeth on the
rear sprocket.
 Different gears are used for
different circumstances.

11.  Cadence - number of revolutions the crank is
moved per minute
 The average recreational cyclist has a cadence of
60-80 rpm
 The average racing cyclist has a cadence of 80-
120 rpm (Lance Armstrong’s cadence is 120 rpm)
 Gear ratios allow optimal cadence in any terrain
 Elevated ground means more force is required to
move the bike forward

12.  Harder to turn the pedals on elevated ground – lower
cadence
 In order to maintain the same cadence, switch to lower
gear
 We proved earlier that to get a higher acceleration, you
needed a smaller gear ratio.
 F=ma; smaller gear ratio will also result in a higher force
output (per amount of force input).
 Elevated slope: need higher output force, same applied
force to the pedals, same cadence. That means a smaller
gear ratio.
 Downhill slope: need a smaller output force, so we need
a higher gear ratio.
 Higher gear ratio: achieve higher vb while maintaining
the same vapp.

13.  Multi-speed bicycles allow gear selection:
 Downhill = High gear
 Flat ground = Medium gear
 Uphill = Low gear
 You can calculate the equivalent wheel size
using the gear ratio.
 Equivalent wheel size = gear ratio x wheel
diameter.
 Ex. if you had a gear ratio of 4.0 and a wheel
diameter of 27”, then the equivalent wheel size
would be 108”.
 That means that using a gear ratio of 4.0 on a
bike with a wheel diameter of 27” is the same
as riding a direct-drive wheel with a diameter
of 108”.
Penny-Farthing
(direct-drive)

14.  Determine distance traveled from each
turn of pedals using equivalent wheel size.
 Distance traveled = circumference of
equivalent wheel
 For our previous example, the distance
traveled for each turn of the pedals would
be:
 p(108”) = 339.3” or 28.3 ft.
 If it was a direct-drive wheel, each turn of
the pedals would take you:
 p(27”) = 84.8” or 7.1 ft.
 Much greater rpm’s of the bike tires at
maximum cadence

15.  How can a bicycle have multiple gear
ratios available?
 Two types of bicycle gears: hub gears
and derailleur gears.
Derailleur Gear
Hub Gear

16. Derailleur Gears
 A transmission system consisting of a chain,
multiple sprockets, and a mechanism
(derailleur) to shift the chain from one
sprocket to another (more common than hub
gears).
 Move lever on handlebar, changes tension in
derailleur cable.
 Change in tension moves derailleur to one
side or the other moving the chain from one
sprocket to another.
 There are two pulleys: the guide pulley and
the tension pulley. Guide pulley pushes chain
from one sprocket to the other. The tension
pulley maintains tension in the chain.

17. Hub Gears
 Work by epicyclic gearing: outer
casing (which is connected to the
spokes of the wheel) rotates at
different speeds depending on
which gear is chosen.
 gears enclosed in hub, protecting
them.
 Used mostly for utility bicycles

18. Advantages
 Mechanism is enclosed within the hub, so it is not
exposed to dirt or weather. Hub gears need very little
maintenance and are very reliable.
 The gear can be changed when the bike is stationary.
(city traffic-starts/stops)
Disadvantages
 Limited space available in the hub, smaller range of
gears than derailleurs.
 They’re heavier and more expensive than derailleur
gears.

19. Gear Ratios (r2/r1)
4
3.5
3
2.5
2
1.5
1
0.5
0
2.5 3 3.5 4 4.5 5 5.5 6
Rear S proc ke t Radius ( r 1 )
High
Medium
Low
Linear (High)
Linear (Medium)
Linear (Low)
Rear Sprocket Radius (r1)
(cm)
Gear Ratios

20. Force Ratios (output over input) For Each Gear Combination
0.5
0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
2.5 3 3.5 4 4.5 5 5.5 6
Re ar S proc ke t Radius ( r 1 )
/
High
Medium
Low
Linear (High)
Linear (Medium)
Linear (Low)
Rear Sprocket Radius (r1)
(cm)
F2/Fapp (N)

21. Calculating the acceleration for the bike
 Measured out 22 m
 Started from rest and timed how long it took to go 22 m.
Motion equation:
Dx = v0t + ½(at2)
 v0 = 0
 a = 2Dx/t2

22. Acceleration For Each Gear Ratio
2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
2.5 3 3.5 4 4.5 5 5.5 6
Rear S proc ke t Radius ( r 1 )
High
Medium
Low
Linear (High)
Linear (Medium)
Linear (Low)
Acceleration (m/s2)
Rear Sprocket Radius (r1)
(cm)

23. Stability
 Prior to the work of David E. H. Jones and
his famous “unridable bicycles”, there were
two main theories on why bicycles are so
stable.
 Front fork swivel and skill of rider
 Gyroscopic effects
 Jones found faults in both theories, and
proposed his own theory, which is now widely
accepted by physicists.

24.  First theory: ridability depends on the freedom of the
front forks to swivel and the skill of the rider.
 Falling bicycle can be saved by proper steering of the
front wheel.
 Bike is easier to ride the faster it is moving - a smaller
steering change is needed to create the centrifugal
correction
 Centrifuge – inertia of bike and friction of tires cause
torque
 Stationary bike impossible to balance
 Faults: when you push a bike it will travel a long time
riderless before falling over (as opposed to the second or
two it takes to fall over when it is stationary).

25.  Second theory: gyroscopic effects from front wheel.
 The angular momentum of the wheel is L=Iw
 L is in same direction as the angular velocity, w, which
equals v x r.
 As long as the bicycle is going straight, the angular
momentum stays in the same position.

26.  If the rider leans left, a torque will be produced which causes a
counterclockwise precession of the bicycle wheel, tending to turn the
bicycle to the left. (Demo)
 t = r x F
 The torque produced by leaning left points to the rear of the bike.
 Torque is always perpendicular to angular momentum (L), and it causes
the angular momentum to change in the direction of the torque vector.
 Angular momentum is in the same direction as the axis of rotation for
the wheel, so if the direction of L shifts backwards, so does the axis of
rotation, turning the wheel left (Demo – volunteer)

27.  David E. H. Jones: non-gyroscopic
bicycle (URB 1).
 second wheel on front fork (clear of
the ground) - spun in opposite
direction of real front wheel, opposing
the gyroscopic effect.
 Fell quickly when pushed rider less
 Still able to ride it (above photo).
 Proved gyroscopic effects of the front wheel weren’t a
significant contributor to the stability of the bicycle (when
rider is present).
 Mass of the wheel is so much less than the mass of the
bicycle and rider that the gyroscopic effects are too small to
really matter (weight of racing bike =15 lb)
 This effect is more important at higher speeds (motorcycles)

28.  So what causes the bicycle to be so stable?
 Trail.
 If you push a bike backwards it will fall over – the two
wheels follow separate paths.
 If you push the bike forwards, the two wheels follow the
same path.
 Trailing frame and back wheel will swing into line
behind the front wheel.
 It isn’t the front wheel straightening out, it’s the back of
the bike swinging into line.

29.  Caster wheels on a shopping cart. If
you push the cart, the forks of the
wheels flip around backwards, and
the wheels “trail” the cart.
 A bicycle’s fork always points forward.
 bicycle wheel’s contact point with the
ground trails the steering axis
intersection with the floor (like caster).
 Pivot point is in front of the place where
the tire touches the ground, the frictional
drag of the tire tends to keep the wheel
straight ahead (like the caster wheel).
 Pull bike backwards, the wheel wants to
turn around

30.  The difference in the wheel contact
point and the steering axis is referred to
as the fork trail of the bicycle,
-determines bike’s stability.
 Larger fork trail = more stable bike.
 If the bicycle leans to the right, the
normal force of the road on the bicycle
tire moves to the left side of the bicycle
and deviates from the steering axis. This
turns the wheel to the right, steering the
bicycle to the side that is required to
keep the bicycle from falling over.
 The greater the fork trail, the greater this
effect.

31.  Since bicycles are balanced on two wheels in a line in
an unstable equilibrium, they must start to fall to one
side or the other.
 A bicycle is stable when it automatically tends to
correct for this unwanted lean.
 Since a larger trail means that the front wheel will turn
more sharply to correct a lean faster, then a larger trail
means that the bicycle is more stable.

32.  Jones proved his trail theory with two
of his “unridable bikes” (URB III and
the URB IV)
 URB III had the fork turned around,
(trail much larger).
 URB III was the most stable bicycle
that Jones tested, actively righting
itself even without a rider.
 Overcorrected the lean at each weave
until it ran out of speed and collapsed.
 Traveled much farther, rider less, than
the same bike with the fork forward
(less trail).
URB III

33.  The URB IV was the most unstable
bicycle that Jones created.
 He moved the front wheel four inches
ahead of its normal position, creating a
negative trail.
 The bicycle fell over immediately
when released rider less, and Jones
described it as “very dodgy to ride.”
 Why is negative trail less stable?
 Demo
URB IV

34.  So if making the trail larger makes the bike more stable,
why don’t all bikes have a large trail?
 Large trail = hard to maneuver.
 A bike that is too stable is sluggish to turn and respond to
the rider’s movements.
 Great for going straight, but turning is difficult
 Most modern bikes have relatively small trails, making
them less stable but more responsive.
Mountain bike - notice
the large fork rake,
making the trail small.

35.  We know trail affects stability
 How do you change trail (T)?
 There are three components to trail:
 q = head angle, R = fork rake,
r = radius of wheel
 Head angle and wheel radius are
usually fixed
 To change Trail, change fork rake
 Bigger fork rake = smaller trail

36. Why Study Bicycles?
 When buying a bicycle, you will have to answer the
following questions:
 Do I want a large or small fork rake?
 Do I want hub gears or derailleur gears?
 What combinations of gear ratios do I want?
 Become an informed shopper.
 Design the bike that is best for your needs.

37.  The Wright Brothers’ pre-aeronautical profession was
bicycle repair and manufacture.
 They used a number of concepts they had learned from
bicycles when they were building their plane:
 The central importance of balance and control.
 The need for strong but lightweight structures.
 The chain-and-sprocket transmission system for propulsion.
 Concerns regarding wind resistance and aerodynamic shape
of the operator

38.  More than $3.5 million was up for grabs this
year at the tour de France, not including the
millions of dollars in endorsements.
 Every rider had a dedicated team trying to
create the “best” bicycle.
 Armstrong used different parts for different
days, depending on the terrain.
 Changed out his gears and wheels each day
to fit the terrain for the day
 Physics led to his 6th tour de France victory
 Would have lost riding the 1885 “Rover”

39. Sources
 Forester, John; “Report on Stability of the Da Hon Bicycle”.
 Jones, David E. H. (1970). The Stability of the Bicycle
 Leisegang, L. and Lee, L. R. Dynamics of a bicycle;
nongyroscopic aspects; American Journal of Physics --
February 1978 -- Volume 46, Issue 2, pp. 130-132
 Peterson, Dr. Randolph
 Popular Mechanics; “Wheels of Fortune”; Wendy Booher
 http://en.wikipedia.org/wiki/Bicycle
 http://www.segway.com/segway/how_it_works.html
 http://hyperphysics.phy-astr.
gsu.edu/hbase/mechanics/bicycle.html#c3
 Zinn, Lennard; Zinn’s Cycling Primer; Block 40.