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Pascal’s Law & the Mechanics of ‘Liquid Levers’: How Enclosed Fluids Boost Applied Force

Introduction

Ever wondered why you can lift a heavy car with the force of your bare hands? With some help from a hydraulic jack, of course?

Also called Principle of Transmission of Fluid Pressure or Pascal’s Principle, the Pascal’s Law states that when pressure is applied to an enclosed fluid, it is transmitted equally to all parts of the fluid as well as to the container walls (Mobley 194).

It is precisely this characteristic of enclosed fluids which makes them capable of multiplying the applied force (Hodanbosi). Such action is similar to liquid levers.

Seventeenth century genius Blaise Pascal coined this superbly useful concept. The syringe and hydraulic press are Blaise Pascal inventions.

Derivation of Mechanical Proof of Pascal’s Law

Figure 1. Pascal’s Law and Force Multiplication
Figure 1. Pascal’s Law and Force Multiplication

Now, pressure is the force acting on unit area. Imagine a container with two ends and housing a fluid as shown in figure 1. Piston at the left end has smaller area (a) than the one at the right end (A). If you apply force (f) on the little piston, you get greater force (F) at the larger piston.

Such is the case because the same pressure (p) acts throughout the enclosed fluid and force rises with increasing area.

Because p= constant = (force / area), we have:

p = f/a = F/A       (1)

F = f * (A / a)       (2)

Since A is greater than a, F is greater than f.

When you apply force on the tiny piston in a hydraulic jack, it gets multiplied at the huge piston and that is precisely how you lift a car with the force of your hands.

Associated Questions

Q.1. Does Pascal’s Law Violate the Principle of Conservation of Energy?

A. No.

Enclosed fluids multiply force, they do not multiply work. Let us illustrate the point through mathematical equations.

In figure 1, the volume of enclosed fluid displaced at the left end will be same as the one displaced at the right end. If h and H are respectively the heights of fluid column displaced at the small and large piston ends:

a * h = A * H                (3)

h = H * (A / a)         (4)

Since A is larger than a, h is greater than H, meaning you have to move the smaller piston through greater distance to obtain little movement of the larger piston.  You must have noticed the same while operating a hydraulic jack.

Now, work input at small piston:

w = f * h                       (5)

And, work output at large piston:

W = F * H                        (6)

Using the expression for h from equation 4 in equation 5, work input at small piston:

w = f * (A / a) * H              (7)

But from equation 2, we have:

F = f * (A / a)

Therefore, from equation 2 and 7, we have:

w = F * H = W           (8)

Work Input = Work Output

Of course, this is an ideal case scenario where we have assumed no friction and zero transmission losses. In the real world, there is a bit of both, making work output somewhat smaller than work input.

Q.2. Mention of Pascal’s Law’s Applications in Hydraulics.

A. Applications of Pascal’s Law:

  • Hydraulic Press: uses hydraulic cylinder for amplifying compressive force through a machine press.
  • Hydraulic Jack: to raise heavy objects such as cars to small height (Physicsabout.com).
  • Hydraulic Brakes: transfer pressure applied on the brake pedal to the brake shoes in all the vehicle’s wheels (Physicsabout.com).
  • Hydraulic Lift: lifts heavy objects to large heights (Physicsabout.com).
  • Power Steering: produces greater turning of vehicle wheels with lesser turning of the steering wheel (Steiner).

Q.3. What is Mechanical Advantage?

A. Mechanical Advantage is the ratio of output force to input force. In other words, mechanical advantage is the force multiplication achieved by a machine (ck-12).

For the mechanism in figure 1:

Mechanical Advantage = (F / f)

Q.4. Name Simple Force Multiplication Devices.

A. There are six simple machines. Except the pulley, all these machines act on the same principle i.e. lower force acting via greater distance creates higher force acting via lesser distance. Six simple machines are:

  • Simple Lever: tiny force applied at greater distance from the fulcrum (also called pivot) can move larger force (load) placed at lesser distance from the fulcrum.
Force Multiplication
Figure 2. Simple Lever & Force Multiplication
Image Courtesy of CR at Spanish Wikipedia at https://en.wikipedia.org/wiki/File:Palanca-ejemplo.jpg

Mechanical Advantage for fig. 2 = (Load Lifted / Effort Applied) = (100 / 5) = 20

  • Wheel and Axle: both rotate together and minor force applied to rotate the large-diameter wheel lifts major load attached to the small-diameter axle.
Wheel-Axle
Figure 3. Wheel-Axle & Mechanical Advantage
Image Courtesy of George Payn Quackenbos at https://en.wikipedia.org/wiki/File:Wheelaxle_quackenbos.gif
  • Pulley: more angle of wrap around the pulley delivers higher mechanical advantage. T2 is always less than T1.

 

Pulley and Force Multiplication
Figure 4. Pulley & Force Multiplication

T1 = T2 * eµθ

Mechanical Advantage = (T1 / T2) = (1 / eµθ)

Where:

T1 is the load or weight being lifted

T2 is the force required to lift the load

e is 2.71828

θ is angle or wrap in radians

µ is coefficient of friction between rope and pulley

  • Inclined Plane: is a ramp that allows you to lift a load by sliding it along instead of raising it vertically.
Inclined Plane
Figure 5. Inclined Plane & Mechanical Advantage

Fi = Fw * sin θ

Where:

Fw is weight of the object

Fi is the force needed to move the object along the inclined plane

Mechanical Advantage = (Fw / Fi) = (1 / sin θ) = always more than 1 because maximum value of sin is 1.

  • Wedge: is equivalent to two inclined planes and is employed to raise heavy bodies from surfaces. Separating devices such as axes, scissors, knives, and saws are wedges and so are holding devices such as nails, staples, and doorstops (Idaho Public Television).
Wedge
Figure 6. Wedge & Force Multiplication

Mechanical Advantage = (L / t) = always greater than 1.

Where:

L is the distance moved by the wedge inside the two surfaces

t is the distance by which the wedge has separated the two surfaces (Georgia State University)

  • Screw: is similar to a long inclined plane woven around a shaft. When using a lever of length L to turn the screw around through one rotation, the load will rise by an amount equal to the pitch of the screw (P).

Mechanical Advantage = (2πL / P)

Screw
Figure 7. Screw & Mechanical Advantage
Conclusion

Pascal’s Law is fundamental to devices using enclosed fluids and is utilized in a whole range of applications. Proper understanding of the principle is essential for developing a firm grasp over the basics of engineering and fluid mechanics.

Works Cited

ck-12. “Mechanical Advantage.” n.d., https://www.ck12.org/physics/mechanical-advantage/lesson/Mechanical-Advantage-MS-PS/. Accessed 20 Mar. 2019.

Georgia State University. “Simple Machines.” N.d., http://hyperphysics.phy-astr.gsu.edu/hbase/Mechanics/incline.html. Accessed 21 Mar. 2019.

Hodanbosi, Carol. “Pascal’s Principle and Hydraulics.” National Aeronautics and Space Administration, n.d., https://www.grc.nasa.gov/www/k-12/WindTunnel/Activities/Pascals_principle.html. Accessed 19 Mar. 2019.

Idaho Public Television. “Simple Machine: Facts.” 2019, http://idahoptv.org/sciencetrek/topics/simple_machines/facts.cfm. Accessed 21 Mar. 2019.

Mobley, Keith. Fluid Power Dynamics. Butterworth-Heinemann, 1999. ScienceDirect, https://www.sciencedirect.com/science/article/pii/B9780750671743500640. Accessed 20 Mar. 2019.

Physicsabout.com. “What is Pascal’s Law and How do we Use it?” Pascal’s Law and its Applications, 16 Jul. 2018, https://physicsabout.com/pascal-law/. Accessed 20 Mar. 2019.

Steiner. “Pascal’s Law and Fluid Automation.” 27 Apr. 2015, https://blog.steinerelectric.com/2015/04/fluid-automation/. Accessed 21 Mar. 2019.

Indrajeetsinh Yadav @ Falcon Words is the author of this paper. For great Engineering Content and Academic Assistance with equations and calculations, write to us at info@falconwords.com or call us at +91-9822052945.

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