Mechanical advantage is a measure of the force amplification achieved by using a tool, mechanical device or machine system. Ideally, the device preserves the input power and simply trades off forces against movement to obtain a desired amplification in the output force. The model for this is the law of the lever. Machine components designed to manage forces and movement in this way are called mechanisms. An ideal mechanism transmits power without adding to or subtracting from it. This means the ideal mechanism does not include a power source, and is frictionless and constructed from rigid bodies that do not deflect or wear. The performance of a real system relative to this ideal is expressed in terms of efficiency factors that take into account friction, deformation and wear.
Contents

The law of the lever 1

Speed ratio 2

Gear trains 3

Chain and belt drives 4

Example: bicycle chain drive 4.1

Block and tackle 5

Efficiency 6

Ideal mechanical advantage 6.1

Actual mechanical advantage 6.2

See also 7

References 8

External links 9
The law of the lever
The lever is a movable bar that pivots on a fulcrum attached to or positioned on or across a fixed point. The lever operates by applying forces at different distances from the fulcrum, or pivot.
As the lever pivots on the fulcrum, points farther from this pivot move faster than points closer to the pivot. The power into and out of the lever must be the same. Power is the product of force and velocity, so forces applied to points farther from the pivot must be less than when applied to points closer in.^{[1]}
If a and b are distances from the fulcrum to points A and B and if force F_{A} applied to A is the input force and F_{B} exerted at B is the output, the ratio of the velocities of points A and B is given by a/b, so the ratio of the output force to the input force, or mechanical advantage, is given by

MA = \frac{F_B}{F_A} = \frac{a}{b}.
This is the law of the lever, which was proven by Archimedes using geometric reasoning.^{[2]} It shows that if the distance a from the fulcrum to where the input force is applied (point A) is greater than the distance b from fulcrum to where the output force is applied (point B), then the lever amplifies the input force. If the distance from the fulcrum to the input force is less than from the fulcrum to the output force, then the lever reduces the input force. Recognizing the profound implications and practicalities of the law of the lever, Archimedes has been famously attributed with the quotation "Give me a place to stand and with a lever I will move the whole world."^{[3]}
The use of velocity in the static analysis of a lever is an application of the principle of virtual work.
Speed ratio
The requirement for power input to an ideal mechanism to equal power output provides a simple way to compute mechanical advantage from the inputoutput speed ratio of the system.
The power input to a gear train with a torque T_{A} applied to the drive pulley which rotates at an angular velocity of ω_{A} is P=T_{A}ω_{A}.
Because the power flow is constant, the torque T_{B} and angular velocity ω_{B} of the output gear must satisfy the relation

P = T_A \omega_A = T_B \omega_B, \!
which yields

MA = \frac{T_B}{T_A} = \frac{\omega_A}{\omega_B}.
This shows that for an ideal mechanism the inputoutput speed ratio equals the mechanical advantage of the system. This applies to all mechanical systems ranging from robots to linkages.
Gear trains
Gear teeth are designed so that the number of teeth on a gear is proportional to the radius of its pitch circle, and so that the pitch circles of meshing gears roll on each other without slipping. The speed ratio for a pair of meshing gears can be computed from ratio of the radii of the pitch circles and the ratio of the number of teeth on each gear, its gear ratio.
Two meshing gears transmit rotational motion.
The velocity v of the point of contact on the pitch circles is the same on both gears, and is given by

v = r_A \omega_A = r_B \omega_B,\!
where input gear A has radius r_{A} and meshes with output gear B of radius r_{B}, therefore,

\frac{\omega_A}{\omega_B} = \frac{r_B}{r_A} = \frac{N_B}{N_A}.
where N_{A} is the number of teeth on the input gear and N_{B} is the number of teeth on the output gear.
The mechanical advantage of a pair of meshing gears for which the input gear has N_{A} teeth and the output gear has N_{B} teeth is given by

MA = \frac{r_B}{r_A} = \frac{N_B}{N_A}.
This shows that if the output gear G_{B} has more teeth than the input gear G_{A}, then the gear train amplifies the input torque. And, if the output gear has fewer teeth than the input gear, then the gear train reduces the input torque.
If the output gear of a gear train rotates more slowly than the input gear, then the gear train is called a speed reducer. In this case, because the output gear must have more teeth than the input gear, the speed reducer will amplify the input torque.
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:

v = r_A \omega_A = r_B \omega_B,\!
where the input sprocket or pulley A meshes with the chain or belt along the pitch radius r_{A} and the output sprocket or pulley B meshes with this chain or belt along the pitch radius r_{B},
therefore

\frac{\omega_A}{\omega_B} = \frac{r_B}{r_A} = \frac{N_B}{N_A}.
where N_{A} is the number of teeth on the input sprocket and N_{B} 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_{A} teeth and the output sprocket has N_{B} teeth is given by

MA = \frac{T_B}{T_A} = \frac{N_B}{N_A}.
The mechanical advantage for friction belt drives is given by

MA = \frac{T_B}{T_A} = \frac{r_B}{r_A}.
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
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 18speed bicycle with 7 in (radius) cranks and 26 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

MA = \frac{F_B}{F_A} = \frac{7}{13} = 0.54.
Now, consider the small and large front sprockets which have 28 and 52 teeth respectively, and consider the small and large rear sprockets which have 16 and 32 teeth each. Using these numbers we can compute the following speed ratios between the front and rear sprockets
Speed ratios

input (small)

input (large)

output (small)

output (large)

speed ratio

crankwheel ratio

total MA

low speed

28

45

19

32

1.14

0.54

0.62

mid 1

19

52

29

32

0.62

0.54

0.33

mid 2

28

39

16

26

0.57

0.54

0.31

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 and the crankwheel 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 backwarddirected reaction force on the ground is indicated). This low mechanical advantage keeps the pedal crank speed low relative to the speed of the drive wheel, even in low gears.
Block and tackle
A block and tackle is an assembly of a rope and pulleys that is used to lift loads. A number of pulleys are assembled together to form the blocks, one that is fixed and one that moves with the load. The rope is threaded through the pulleys to provide mechanical advantage that amplifies that force applied to the rope.^{[4]}
In order to determine the mechanical advantage of a block and tackle system consider the simple case of a gun tackle, which has a single mounted, or fixed, pulley and a single movable pulley. The rope is threaded around the fixed block and falls down to the moving block where it is threaded around the pulley and brought back up to be knotted to the fixed block.
The mechanical advantage of a block and tackle equals the number of sections of rope that support the moving block; shown here it is 2, 3, 4, 5, and 6, respectively.
Let S be the distance from the axle of the fixed block to the end of the rope, which is A where the input force is applied. Let R be the distance from the axle of the fixed block to the axle of the moving block, which is B where the load is applied.
The total length of the rope L can be written as

L = 2R + S + K, \!
where K is the constant length of rope that passes over the pulleys and does not change as the block and tackle moves.
The velocities V_{A} and V_{B} of the points A and B are related by the constant length of the rope, that is

\dot{L}=2\dot{R} + \dot{S}=0,
or

\dot{S} = 2\dot{R}.
The negative sign shows that the velocity of the load is opposite to the velocity of the applied force, which means as we pull down on the rope the load moves up.
Let V_{A} be positive downwards and V_{B} be positive upwards, so this relationship can be written as the speed ratio

\frac{V_A}{V_B} = \frac{\dot{S}}{\dot{R}} = 2,
where 2 is the number of rope sections supporting the moving block.
Let F_{A} be the input force applied at A the end of the rope, and let F_{B} be the force at B on the moving block. Like the velocities F_{A} is directed downwards and F_{B} is directed upwards.
For an ideal block and tackle system there is no friction in the pulleys and no deflection or wear in the rope, which means the power input by the applied force F_{A}V_{A} must equal the power out acting on the load F_{B}V_{B}, that is

F_A V_A = F_B V_B.\!
The ratio of the output force to the input force is the mechanical advantage of an ideal gun tackle system,

MA = \frac{F_B}{F_A} = \frac{V_A}{V_B} = 2.\!
This analysis generalizes to an ideal block and tackle with a moving block supported by n rope sections,

MA = \frac{F_B}{F_A} = \frac{V_A}{V_B} = n.\!
This shows that the force exerted by an ideal block and tackle is n times the input force, where n is the number of sections of rope that support the moving block.
Efficiency
Mechanical advantage that is computed using the assumption that no power is lost through deflection, friction and wear of a machine is the maximum performance that can be achieved. For this reason, it is often called the ideal mechanical advantage (IMA). In operation, deflection, friction and wear will reduce the mechanical advantage. The amount of this reduction from the ideal to the actual mechanical advantage (AMA) is defined by a factor called efficiency, a quantity which is determined by experimentation.
As an ideal example, using a block and tackle with six ropes and a 600 pound load, the operator would be required to pull the rope six feet and exert 100 pounds of force to lift the load one foot. Both the ratios F_{out} / F_{in} and V_{in} / V_{out} from below show that the IMA is six. For the first ratio, 100 pounds of force in results in 600 pounds of force out; in the real world, the force out would be less than 600 pounds. The second ratio also yields a MA of 6 in the ideal case but fails in real world calculations; it does not properly account for energy losses. Subtracting those losses from the IMA or using the first ratio yields the AMA. The ratio of AMA to IMA is the mechanical efficiency of the system.
Ideal mechanical advantage
The ideal mechanical advantage (IMA), or theoretical mechanical advantage, is the mechanical advantage of a device with the assumption that its components do not flex, there is no friction, and there is no wear. It is calculated using the physical dimensions of the device and defines the maximum performance the device can achieve.
The assumptions of an ideal machine are equivalent to the requirement that the machine does not store or dissipate energy; the power into the machine thus equals the power out. Therefore, the power P is constant through the machine and force times velocity into the machine equals the force times velocity outthat is,

P = F_{in}v_{in}= F_{out}v_{out}.
The ideal mechanical advantage is the ratio of the force, or effort, out of the machine relative to the force or effort into the machine, or

IMA = \frac {F_{out}} {F_{in}}.
Applying the constant power relationship yields a formula for this ideal mechanical advantage in terms of the speed ratio:

IMA = \frac {F_{out}} {F_{in}} = \frac {v_{in}} {v_{out}}.
The speed ratio of a machine can be calculated from its physical dimensions. The assumption of constant power thus allows use of the speed ratio to determine the maximum value for the mechanical advantage.
Actual mechanical advantage
The actual mechanical advantage (AMA) is the mechanical advantage determined by physical measurement of the input and output forces. Actual mechanical advantage takes into account energy loss due to deflection, friction, and wear.
The AMA of a machine is calculated as the ratio of the measured force output to the measured force input,

AMA = \frac {F_{out}} {F_{in}},
where the input and output forces are determined experimentally.
The ratio of the experimentally determined mechanical advantage to the ideal mechanical advantage is the efficiency η of the machine,

\eta =\frac{AMA}{IMA}.
See also
References

^ J. J. Uicker, G. R. Pennock, and J. E. Shigley, 2003, Theory of Machines and Mechanisms, Oxford University Press, New York.

^

^ John Tzetzes Book of Histories (Chiliades) 2 p 129130, 12th century AD, translation by Francis R. Walton

^ Ned Pelger, ConstructionKnowledge.net

Fisher, Len (2003), How to Dunk a Doughnut: The Science of Everyday Life, Arcade Publishing, .

United States Bureau of Naval Personnel (1971), Basic machines and how they work (Revised 1994 ed.), Courier Dover Publications, .
External links

Gears and pulleys

Mechanical engineering — pulleys

Mechanical advantage — video
This article was sourced from Creative Commons AttributionShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and USA.gov, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for USA.gov and content contributors is made possible from the U.S. Congress, EGovernment Act of 2002.
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a nonprofit organization.