Editing Mechanical advantage (section)

== Chain and belt drives ==
Mechanisms consisting of two sprockets connected by a chain, or two pulleys connected by a belt are designed to provide a specific mechanical advantage in power transmission systems.

The velocity ''v'' of the chain or belt is the same when in contact with the two sprockets or pulleys:
:<math> v = r_A \omega_A = r_B \omega_B,\!</math>
where the input sprocket or pulley ''A'' meshes with the chain or belt along the pitch radius ''r<sub>A</sub>'' and the output sprocket or pulley ''B'' meshes with this chain or belt along the pitch radius ''r<sub>B</sub>'',

therefore
:<math> \frac{\omega_A}{\omega_B} = \frac{r_B}{r_A} = \frac{N_B}{N_A}.</math>
where ''N<sub>A</sub>'' is the number of teeth on the input sprocket and ''N<sub>B</sub>'' is the number of teeth on the output sprocket.  For a [[toothed belt]] drive, the number of teeth on the sprocket can be used.  For friction belt drives the pitch radius of the input and output pulleys must be used.

The mechanical advantage of a pair of a chain drive or toothed belt drive with an input sprocket with  ''N<sub>A</sub>''  teeth and the output sprocket has ''N<sub>B</sub>'' teeth is given by
:<math> \mathit{MA} = \frac{T_B}{T_A}  = \frac{N_B}{N_A}.</math>

The mechanical advantage for friction belt drives is given by
:<math> \mathit{MA} = \frac{T_B}{T_A} = \frac{r_B}{r_A}.</math>

Chains and belts dissipate power through friction,  stretch and wear, which means the power output is actually less than the power input, which means the mechanical advantage of the real system will be less than that calculated for an ideal mechanism.  A chain or belt drive can lose as much as 5% of the power through the system in friction heat, deformation and wear, in which case the efficiency of the drive is 95%.

=== Example: bicycle chain drive ===
[[Image:Bicycle mechanical advantage.svg|thumb|500px|center|Mechanical advantage in different gears of a bicycle. Typical forces applied to the bicycle pedal and to the ground are shown, as are corresponding distances moved by the pedal and rotated by the wheel. Note that even in low gear the MA of a bicycle is less than 1.]]

Consider the 18-speed bicycle with 7&nbsp;in (radius) cranks and 26&nbsp;in (diameter) wheels.  If the sprockets at the crank and at the rear drive wheel are the same size, then the ratio of the output force on the tire to the input force on the pedal can be calculated from the law of the lever to be
:<math> \mathit{MA} = \frac{F_B}{F_A} = \frac{7}{13} = 0.54.</math>

Now, assume that the front sprockets have a choice of 28 and 52 teeth, and that the rear sprockets have a choice of 16 and 32 teeth.  Using different combinations, we can compute the following speed ratios between the front and rear sprockets

{| class="wikitable" style="text-align: center; width: 500px;"
|+  Speed ratios and total MA
|-
! scope="col"|
! scope="col"| input
! scope="col"| output
! scope="col"| speed ratio
! scope="col"| crank-wheel ratio
! scope="col"| total MA
|-
! scope="row" | low speed
| 28 || 32 || 1.14 || 0.54 || 0.62
|-
! scope="row" | mid 1
| 52 || 32 || 0.62 || 0.54 || 0.33
|-
! scope="row" | mid 2
| 28 || 16 || 0.57 || 0.54 || 0.31
|-
! scope="row" | high speed
| 52 || 16 || 0.30 || 0.54 || 0.16
|}

The ratio of the force driving the bicycle to the force on the pedal, which is the total mechanical advantage of the bicycle, is the product of the speed ratio (or teeth ratio of output sprocket/input sprocket) and the crank-wheel lever ratio.

Notice that in every case the force on the pedals is greater than the force driving the bicycle forward (in the illustration above, the corresponding backward-directed reaction force on the ground is indicated).