Bicycle Gears and Energy Worksheet
Fill in the table below using measurements from Autodesk®Inventor® and math equations that
represent the bike, as reviewed in the presentation. First gather the info for the initial setup in the
Inventor assembly, then for the changed setup with a larger rear gear.
Initial Setup
44
17
44/17 = 2.59
Changed Setup
44
27
44/27 = 1.63
Higher or lower gear?
Angle rear wheel rotates for full
rotation of front cranks
Rear wheel angle / front gear
angle
Higher
932 degrees = 360° + 360° +
180° + 32°
932° / 360° = 2.59
Lower
587° = 360° + 180° + 47°
Diameter of wheel
Radius of wheel
Circumference of wheel
d = 690mm = 0.69m
r = 0.69m / 2 = 0.35m
C = π * d = 2.17m
C * g.r. = 2.17m * 2.59 = 5.6m
0.69m
0.35m
C = π * d = 2.17m
= 2.17m * 1.63 = 3.5m
L = 170mm = 0.17m
(0.35m / 0.17m) * 2.59 = 5.33
L = 0.17m
(0.35m / 0.17m) * 1.63 = 3.36
Distance pedal travels in one
pedal rotation
Distance rear wheel travels for
one pedal rotation
C = π * 2 * L = 1.07m
1.07m
C * G.R. = 1.07m * 5.33 = 5.7m
1.07m * 3.36 = 3.6m
Mechanical Advantage
1 / 5.18 = 0.19
1 / 3.36 = 0.30
Output force for 1000N input
force
Fout = 1000N * 0.19 = 190N
Fout = 1000N * 0.30 = 300N
# teeth front gear
# teeth rear gear
Gear ratio
Meters of development
Crank length
Gain ratio
587° / 360° = 1.63
Additional Questions:
Many people simply give the size of the front and rear gears when describing the gearing on their
bicycle. Suppose there are two different bicycles, one with a gearing of 52/20 and another with 42/17.
All other things being equal, which bicycle would travel farther for one full rotation of the cranks? Why?
The 52/20 bicycle (gear ratio = 2.6) would travel farther than the 42/17 bike (gear ratio = 2.47)
because it is a higher gear, which means more distance for each pedal, provide the wheels are the
same.
Autodesk® Digital STEAM Workshop – Worksheet Answers – Bicycle Gears and Energy
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Which of these bicycles, the 52/20 or 42/17, would feel easier to pedal? Why?
The 42/17 setup would feel easier to pedal because it has a larger mechanical advantage, given that it
is the lower gear for the same bicycle. That is, the force out will be greater in lower gear (but the
distance output will be smaller).
How can you increase the mechanical advantage of a bicycle?
Mechanical advantage is 1 / gain ratio, so DECREASING the gain ratio increases the mechanical
advantage. To decrease gain ratio, you can INCREASE the wheel radius, DECREASE the crank length, or
DECREASE the gear ratio (which itself means either DECREASE the number of front teeth or INCREASE
the number of rear teeth). The point here is more to trace how variables affect other quantities, and
to gain some comfort with fractions.
Explain the relationship between force and distance in the output of a bicycle in terms of energy.
For a given energy input, an INCREASE in the output force will correspond to a DECREASE in output
distance, vice versa.
Going Further:
Try changing the front and rear gears on the bicycle in Inventor. What’s the biggest mechanical
advantage you can generate?
Students can play around to see what will fit onto the bicycle, and what other parts fail with each
change.
Useful Equations:
gear ratio = # teeth on front gear / # teeth on rear gear
g.r. = Nfront / Nrear
Circumference = π x diameter
C=πxd
diameter = 2 x radius
d=2xr
Meters of development = Circumference x gear ratio
MoD = C x g.r.
Gain Ratio = (wheel radius / crank length) x gear ratio
G.R. = (rwheel / Lcrank) x g.r.
Mechanical Advantage = 1 / gain ratio = Force out / Force in
M.A. = 1 / G.R. = Fout / Fin
Autodesk® Digital STEAM Workshop – Worksheet Answers – Bicycle Gears and Energy
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