MECHANICAL ADVANTAGE: FORMULA, EQUATIONS, CALCULATION AND EXAMPLES - PHYSICAL - 2024

The mechanical advantage is the dimensionless factor that quantifies the ability of a mechanism to amplify disminuir- in some cases the force is exerted through it. The concept applies to any mechanism: from a pair of scissors to a sports car engine.

The idea is for machinery to transform the force that the user applies on it into a much greater force that represents profit, or to reduce it to carry out a delicate task.

Figure 1. The hydraulic lift is a machine with a mechanical advantage greater than 1. Source: Pixabay.

It should be borne in mind that when a mechanism is actuated, a part of the applied force is inevitably invested in counteracting friction. Therefore the mechanical advantage is classified into actual mechanical advantage and ideal mechanical advantage.

Definition and formulas

The actual mechanical advantage of a machine is defined as the ratio between the magnitude of the force exerted by the machine on the load (output force) and the force required to operate the machine (input force):

Real Mechanical Advantage VMR = Exit Force / Entry Force

While for its part, the ideal mechanical advantage depends on the distance traveled by the input force and the distance traveled by the output force:

Ideal mechanical advantage VMI = Inlet distance / Outlet distance

Being quotients between quantities with the same dimensions, both advantages are dimensionless (without units) and also positive.

In many cases, such as the wheelbarrow and hydraulic press, the mechanical advantage is greater than 1, and in others, the mechanical advantage is less than 1, for example in the fishing rod and grippers.

Ideal mechanical advantage VMI

IMV is related to the mechanical work that is carried out at the entrance and exit of a machine. The input work, which we will call W _i, is broken down into two components:

W _i = Work to overcome friction + Work out

An ideal machine does not need to do work to overcome friction, therefore the work at the input would be the same as at the output, denoted as W _or:

Work on entry = Work on exit → W _i = W _o.

Since in this case work is force times distance, we have: W _i = F _i. yes _i

Where F _i and s _i are the initial force and distance respectively. The output work is expressed analogously:

W _o = F _o. s _or

In this case F _o and s _o are the force and the distance that the machinery delivers, respectively. Now both jobs are matched:

F _i. s _i = F _o. s _or

And the result can be rewritten in the form of quotients of forces and distances:

(s _i / s _o) = (F _o / F _i)

Precisely the distance quotient is the ideal mechanical advantage, according to the definition given at the beginning:

VMI = s _i / s _o

Efficiency or performance of a machine

It is reasonable to think about the efficiency of the transformation between both jobs: the input and the output. Denoting efficiency as e, it is defined as:

e = Output work / Input work = W _o / W _i = F _o. s _o / F _i. yes _i

Efficiency is also known as mechanical performance. In practice, the output work never exceeds the input work because of friction losses, therefore the quotient given by e is no longer equal to 1, but less.

An alternative definition involves power, which is the work done per unit of time:

e = Power output / Power input = P _o / P _i

Real mechanical advantage VMR

The actual mechanical advantage is simply defined as the quotient between the output force F _o and the input force F _i:

VMR = F _o / F _i

Relationship between VMI, VMR and efficiency

The efficiency e can be rewritten in terms of VMI and VMR:

e = F _o. s _o / F _i. s _i = (F _o / F _i). (s _o / s _i) = VMR / VMI

Therefore, the efficiency is the quotient between the real mechanical advantage and the ideal mechanical advantage, the former being less than the latter.

Calculation of VMR knowing the efficiency

In practice, the VMR is calculated by determining the efficiency and knowing the VMI:

VMR = e. VMI

How is mechanical advantage calculated?

The calculation of the mechanical advantage depends on the type of machinery. In some cases it is convenient to carry it out by transmitting forces, but in other types of machines, such as pulleys for example, it is the torque or torque τ that is transmitted.

In this case, the VMI is calculated by equating the moments:

Output torque = Input torque

The magnitude of the torque is τ = Frsen θ. If the force and the position vector are perpendicular, between them there is an angle of 90º and sin θ = sin 90º = 1, obtaining:

F _or. r _o = F _i. r _i

In mechanisms such as the hydraulic press, which consists of two chambers interconnected by a transverse tube and filled with a fluid, pressure can be transmitted through pistons that move freely in each chamber. In that case, the VMI is calculated by:

Outlet pressure = Inlet pressure

Figure 2. Diagram of the hydraulic press. Source: Cuéllar, J. 2015. Physics II. McGraw Hill.

Examples

- Example 1

The lever consists of a thin bar supported by a support called a fulcrum, which can be positioned in various ways. By applying a certain force, called "power force", a much greater force is overcome, which is the load or resistance.

Figure 3. First-class lever. Source: Wikimedia Commons. CR

There are several ways to locate the fulcrum, power force, and load to achieve mechanical advantage. Figure 3 shows the first-class lever, similar to a rocker, with the fulcrum located between the power force and the load.

For example, two people of different weight can balance on the seesaw or go up and down if they sit at appropriate distances from the fulcrum.

To calculate the VMI of the first degree lever, since there is no translation and no friction is considered, but there is rotation, the moments are equalized, knowing that both forces are perpendicular to the bar. Here F _i is the power force and F _o is the load or resistance:

F _or. r _o = F _i. r _i

F _o / F _i = r _i / r _o

By definition VMI = F _o / F _i, then:

VMI = r _i / r _o

In the absence of friction: VMI = VMR. Note that VMI can be greater or less than 1.

- Example 2

The ideal mechanical advantage of the hydraulic press is calculated through the pressure, which according to Pascal's principle, is fully transmitted to all points of the fluid confined in the container.

The input force F ₁ in FIG. 2 is applied to the small piston of area A ₁ on the left, and the output force F ₂ is obtained from the large piston of area A ₂ on the right. So:

Inlet pressure = Outlet pressure

Pressure is defined as force per unit area, therefore:

(F ₁ / A ₁) = (F ₂ / A ₂) → A ₂ / A ₁ = F ₂ / F ₁

Since VMI = F ₂ / F ₁, we have the mechanical advantage through the ratio between the areas:

VMI = A ₂ / A ₁

Since A ₂ > A ₁, the VMI is greater than 1 and the effect of the press is to multiply the force applied to the small piston F ₁.

References

1. Cuéllar, J. 2009. Physics II. 1st. Edition. McGraw Hill.
2. Kane, J. 2007. Physics. 2nd. Edition. Editorial Reverté.
3. Tippens, P. 2011. Physics: Concepts and Applications. 7th Edition. Mcgraw hill
4. Wikipedia. Lever. Recovered from: es.wikipedia.org.
5. Wikipedia. Mechanical advantage. Recovered from: es.wikipedia.org.