Physics and Mathematics
Energy of a Liquid
1. Statement of the Concept
A moving or stationary liquid possesses energy by virtue of its pressure, position, and motion. These three forms — Pressure Energy, Potential Energy, and Kinetic Energy — together determine the total mechanical energy of a liquid element.
2. Explanation of Each Energy Form
(a) Pressure Energy
Pressure energy is the energy a liquid element possesses because of the pressure acting on it.
When a force due to pressure pushes a volume of liquid through a distance, work is done on the liquid.
If a volume [V] is pushed into a tube where the pressure is [P], the work done is:
[\text{Pressure Energy} = PV]
Energy per unit mass:
$\left[\dfrac{PV}{m} = \dfrac{P}{\rho}\right]$
Energy per unit volume:
$\left[P\right]$
(b) Potential Energy
A liquid at height [h] above a reference level possesses gravitational potential energy.
For a mass [m] of liquid:
[\text{Potential Energy} = mgh]
Energy per unit mass:
$\left[gh\right]$
Energy per unit volume (since [m = \rho V]):
$\left[\rho gh\right]$
(c) Kinetic Energy
A moving liquid has kinetic energy due to its velocity [v].
For a mass [m], kinetic energy is:
[\text{Kinetic Energy}] [= \dfrac{1}{2} mv^{2}]
Energy per unit mass:
$\left[\dfrac{1}{2} v^{2}\right]$
Energy per unit volume:
$\left[\dfrac{1}{2} \rho v^{2}\right]$
3. Dimensions and Units
| Quantity | Expression | Dimensions | SI Unit |
|---|---|---|---|
| Pressure Energy | [PV] | ([ML^{2}T^{-2}]) | Joule |
| Potential Energy | [mgh] | ([ML^{2}T^{-2}]) | Joule |
| Kinetic Energy | [\dfrac{1}{2}mv^{2}] | ([ML^{2}T^{-2}]) | Joule |
4. Key Features
- Every liquid element simultaneously possesses pressure, potential, and kinetic energy.
- These energies collectively form the basis of Bernoulli’s equation (link to Bernoulli’s topic when we create it).
- Pressure energy arises from external forces; potential energy arises from the gravitational field; kinetic energy arises from motion.
- All three energies are interchangeable when fluid flows streamline-wise.
5. Important Formulas to Remember
| Energy Type | Energy per Unit Mass | Energy per Unit Volume |
|---|---|---|
| Pressure Energy | [\dfrac{P}{\rho}] | [P] |
| Potential Energy | [gh] | [\rho gh] |
| Kinetic Energy | [\dfrac{1}{2} v^{2}] | [\dfrac{1}{2} \rho v^{2}] |
6. Conceptual Questions with Solutions
1. Why does pressure energy increase when depth increases?
At greater depths, liquid pressure increases due to the weight of the liquid column above. Higher pressure means more work is needed to push a volume of liquid, increasing pressure energy.
2. Does a stationary fluid possess kinetic energy?
No. Kinetic energy requires motion. A stationary liquid has only pressure and potential energy.
3. Why is potential energy proportional to height?
Because gravitational work done in lifting a liquid mass \[m\] through height \[h\] is \[mgh\]. Thus, energy increases linearly with height.
4. Is pressure energy stored inside the liquid?
Yes. It exists because external pressure forces compress and push the fluid. It becomes apparent during flow.
5. Why does kinetic energy increase with velocity squared?
Because work needed to accelerate a fluid mass increases with the square of velocity: \[\dfrac{1}{2} mv^{2}\].
6. Which energy is responsible for lifting water in a fountain?
Kinetic energy of fast-moving water is converted into potential energy as water rises.
7. Can pressure energy convert to kinetic energy?
Yes. In a constricted pipe (like a nozzle), pressure drops and velocity increases — pressure energy converts to kinetic energy.
8. Why is pressure energy considered “work done on the fluid”?
Because work is done by external pressure to force fluid elements to move a certain distance.
9. Why are these energies additive?
Because they represent independent physical processes: pressure work, gravitational work, and motion. Total energy is their sum.
10. Which energy form changes when a liquid flows uphill?
Kinetic energy converts into potential energy.
7. FAQ / Common Misconceptions
1. Does higher pressure mean faster speed?
Not always. In fact, in many flows (like in Bernoulli’s principle), **higher pressure corresponds to lower speed**.
2. Pressure energy is not real energy — is that true?
False. It is real energy associated with work done by pressure forces.
3. Liquid at rest has no energy?
Incorrect. It has pressure energy and potential energy.
4. Kinetic energy depends only on speed, not mass?
Incorrect. Total kinetic energy is \[\dfrac{1}{2}mv^{2}\].
5. Potential energy is only for solids?
No. Any mass in a gravitational field has potential energy, including liquids.
6. Deep water has less potential energy.
No. It depends on the height above reference level, not depth below open surface.
7. Pressure energy cannot change form.
It can. Pressure energy converts into kinetic or potential energy during flow.
8. All energies increase with depth.
Only pressure energy increases with depth; potential and kinetic depend on height and velocity.
9. A liquid moving horizontally has no potential energy.
It does, as long as it has height \[h\] above the reference level.
10. Pressure energy equals force.
No. Force is different; pressure energy is work done due to pressure.
8. Practice Questions with Step-by-Step Solutions
1. Calculate the pressure energy per unit mass of water at pressure [2 \times 10^{5} , \text{Pa}].
Solution:
[\text{Pressure energy per unit mass}] [= \dfrac{P}{\rho}]
For water, [\rho = 1000Â \text{kg/m}^{3}].
$\left[\dfrac{2 \times 10^{5}}{1000} = 200Â \text{J/kg}\right]$
2. Find the potential energy per unit mass of a liquid at height [20 , \text{m}].
Solution:
[gh = 9.8 \times 20 = 196 , \text{J/kg}]
3. A liquid flows at [5 , \text{m/s}]. Compute its kinetic energy per unit mass.
Solution:
[\dfrac{1}{2}v^{2}] [= \dfrac{1}{2} \times 25] [= 12.5Â \text{J/kg}]
4. A 3 kg liquid mass rises by [10 , \text{m}]. Find its increase in potential energy.
Solution:
[mgh] [= 3 \times 9.8 \times 10] [= 294 \text{J}]
5. A fluid’s velocity increases from [2 , \text{m/s}] to [6 , \text{m/s}]. Find the increase in kinetic energy per unit mass.
Solution:
Initial KE per unit mass:
[
\dfrac{1}{2}(2^{2}) = 2
]
Final KE per unit mass:
[
\dfrac{1}{2}(6^{2}) = 18
]
Increase:
[
18 – 2 = 16 , \text{J/kg}
]
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