Analysis of Simple Machines

A machine can be described as an assembly of rigid bodies for transmitting force, motion or work

A machine is usually designed so that a desired load output from the machine is greater than the effort put in, for example a lever can lift a high load with a reasonably small force applied. The ratio of the load to effort is known as the mechanical advantage.

Mechanical Advantage (M.A) = ^{load ( W)} / _{effort (E)}

As the load and the effort are forces and have the same units (N) the mechanical advantage has no units and is simply a ratio of these two values. In practice this mechanical advantage is obtained by allowing the effort to move a greater distance than the load. 

The diagram above shows how a simple lever operates with the effort applied and resulting load. 

It is clear from this diagram that the effort must travel a greater distance than the load to raise it so that the weight is lifted. The greater the length of this level the smaller the effort that needs to be applied as the resulting moment will be greater.

Since the distance that the effort and load moves takes place over the same period of time, the ratio of these distances is also equal to the ratio of the velocities. 

The velocity ratio of a machine is defined as:

Velocity Ratio (V.R) = ^{distance moved by effort} / _{distance moved by load}

Just as the  mechanical advantage, velocity ratio has no units. 

For an ideal machine where no losses are present, such as friction, the velocity ratio is equal to the mechanical advantage. This statement leads us to the principle that the work input into a machine is the same as the work output by the machine.

Work input = effort x distance moved by effort

Work output = load x distance moved by load

Work input = work output

Effort x distance moved by effort = load x distance moved by load 

Which leads to:

^{Distance moved by effort} / _{distance moved by load} = ^load / _effort

For an ideal machine:

V.R = M.A

In practice, the work output is always less than the work input because of friction and other losses. This leads to examining the efficiency of a machine. The efficiency of a machine can be described by the useful work done by the machine divided by the work put into the machine.

Efficiency = ^{work output}/_{work input}

Efficiency = ^{load x distance moved by the load} / _{actual effort x distance moved by the effort}

Which leads to:

Efficiency ( η) = ^M.A / _V.R

Example

The load on a lever is 2 kN and acts 20 mm from the fulcrum. The effort is 200 mm from the fulcrum. Assuming 80% efficiency, calculate the effort applied.

The velocity ratio.

V.R = ^{distance moved by effort} / _{distance moved by load} = ²⁰⁰ / ₂₀ = 10

Efficiency η = ^M.A / _V.R

Rearranging gives:

M.A = η x V.R = 0.8 x 10 = 8

The mechanical advantage given by the lever is 8.

M.A = ^load / _effort

Effort = ^load / _M.A = ²⁰⁰⁰ / ₈ = 250N

Example

A simple pulley system is shown opposite. A 50N effort is used to move an 80N load. The final effort moves 2 metres and at the same time the load moves 1 metre.

What is the Mechanical advantage, velocity ratio and efficiency of the pulley?

M.A = ^load / _effort = ⁸⁰ / ₅₀ = 1.6

V.R = ^{distance moved by effort} / _{distance moved by load} = ² / ₁ = 2

η = ^M.A / _V.R = ^1.6 / ₂ =  0.8 or 80%

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