Conservation of Mechanical Energy: Stepwise Derivation and Solved Numericals
The conservation of mechanical energy is a fundamental concept in physics, especially in mechanics. It states that in the absence of non-conservative forces like air resistance or friction, the total mechanical energy of a system remains constant throughout its motion. Mechanical energy is the sum of kinetic energy (energy due to motion) and potential energy (energy due to position or configuration). This principle allows us to predict the motion and speed of objects in various scenarios without tracking every force in detail.
Understanding Mechanical Energy Conservation
Mechanical energy conservation applies when only conservative forces (like gravity or spring force) act on the object. In these cases, energy merely transforms from one form to another. For example, as a ball falls freely, its gravitational potential energy decreases, while its kinetic energy increases by the same amount. The total mechanical energy, which is the sum of both, stays the same.
Key Formula for Mechanical Energy
| Term | Formula | Description |
|---|---|---|
| Kinetic Energy (K.E.) | K.E. = (1/2)mv2 | Energy due to motion (m = mass, v = velocity) |
| Potential Energy (P.E.) | P.E. = mgh | Gravitational potential energy (m = mass, g = acceleration due to gravity, h = height) |
| Total Mechanical Energy | E = K.E. + P.E. | Sum of kinetic and potential energy |
The law of conservation of mechanical energy can be expressed as:
K.E.initial + P.E.initial = K.E.final + P.E.final
This holds true when non-conservative forces like air resistance and friction are absent or negligible.
Worked Example: Free Fall
A suitcase is dropped from rest at a certain height. If we ignore air resistance, what will its velocity be just before it strikes the ground?
- At the top: K.E. = 0 (at rest), P.E. = mgh
- At the bottom: K.E. = (1/2)mv2, P.E. = 0 (h = 0)
By conservation of mechanical energy:
mgh = (1/2)mv2
The mass, m, cancels out. Solving for v:
v = √(2gh)
If h = 2 m and g = 9.8 m/s2,
v = √(2 × 9.8 × 2) = √39.2 ≈ 6.26 m/s
Experiment: Conversion of Energy
To visualize conservation of mechanical energy, you can perform a simple experiment:
- Lift one end of a pipe to a certain height above a table.
- Measure that height.
- Place a marble at the top and let it roll through the pipe onto the table.
- The marble's potential energy at the top is converted to kinetic energy as it moves.
When the marble leaves the pipe, most of the original potential energy has become kinetic energy.
Further Worked Examples
| Scenario | Process | Key Steps |
|---|---|---|
| Pendulum (No friction) | Potential energy at highest point, kinetic energy at lowest. |
Use conservation: P.E.high = K.E.low. Show that velocity depends on height, not mass. |
| Roller Coaster (Frictionless) | Energy at top converts to kinetic at various points. |
Use conservation to find velocity at the top of loops and bottom of track. m cancels from both sides; calculations depend only on height difference. |
| Inclined Plane | Object starts from rest at a height; slides down plane. |
Calculate potential energy at top, kinetic at bottom. P.E.top = K.E.bottom. |
Potential Energy and Object Motion
Consider a ball dropped from a height of 3 m. Its potential energy relative to the ground is:
P.E. = mgh (with h = 3 m)
As it falls, potential energy decreases while kinetic energy increases by the same amount.
Its velocity just before hitting the ground is obtained via v = √(2gh). This velocity is independent of the ball's mass, provided no friction acts.
Important Observations
- When only conservative forces act, mechanical energy is conserved.
- Calculations often allow for mass (m) to cancel out, making final velocities mass-independent.
- Non-conservative forces (friction, air resistance) reduce mechanical energy by converting it to heat or sound.
Practice and Next Steps
- Practice similar problems using Conservation of Mechanical Energy.
- Explore related concepts with Kinetic Energy and Potential and Kinetic Energy.
- Understand energy conversion and more with Energy Conservation and Work, Energy and Power.
Summary for Revision
| Key Point | Explanation |
|---|---|
| Mechanical Energy | Sum of kinetic and potential energies |
| Conservative Forces | Do not reduce total mechanical energy; enable energy transformation |
| Friction/Non-Conservative Forces | Convert some mechanical energy to heat, so mechanical energy decreases |
| Practical Application | Used to solve velocity, height, or energy problems in free fall, roller coasters, inclined planes, and pendulums |
To master these concepts, practice solving a variety of numerical and conceptual problems. For in-depth explanations and exercises, explore more on Conservation of Mechanical Energy and related pages.
FAQs on Conservation of Mechanical Energy – Definition, Law, and Examples
1. What is conservation of mechanical energy?
Conservation of mechanical energy states that in a system acted upon only by conservative forces, the total mechanical energy (the sum of kinetic energy (K.E.) and potential energy (P.E.)) remains constant. This means energy is neither created nor destroyed, but can be transformed between forms as long as no dissipative (non-conservative) forces, like friction or air resistance, are present.
2. What is the formula for conservation of mechanical energy?
The formula for conservation of mechanical energy is:
Etotal = K.E. + P.E. = constant
This can be written as:
- K.E.initial + P.E.initial = K.E.final + P.E.final
3. Is mechanical energy always conserved?
Mechanical energy is conserved only in the absence of non-conservative forces.
- Conserved: When only conservative forces (like gravity, spring force) are present.
- Not conserved: When non-conservative forces (such as friction or air resistance) do work, mechanical energy is converted to other forms like heat or sound.
4. What are real-life examples of conservation of mechanical energy?
Common real-life examples include:
- A swinging pendulum (neglecting air resistance)
- A body freely falling under gravity
- A roller coaster in a frictionless track
- A spring-mass system without energy loss
5. How is mechanical energy conserved in motion?
Mechanical energy is conserved during motion when no external non-conservative forces act. As an object moves, its kinetic and potential energies may change, but their total sum stays constant. For example, when an object falls, its potential energy decreases while its kinetic energy increases by an equivalent amount.
6. What are the main types of mechanical energy?
Types of mechanical energy:
- Kinetic Energy (K.E.): Energy due to motion, given by ½mv2
- Potential Energy (P.E.): Energy due to position or configuration, such as mgh (gravitational) or ½kx2 (spring)
7. Is mechanical energy conserved in the presence of friction? Why or why not?
No, mechanical energy is NOT conserved when friction is present. This is because friction is a non-conservative force that transforms mechanical energy into heat and other forms, reducing the total mechanical energy of the system. Only the total energy (mechanical + thermal) is conserved, but mechanical energy alone decreases.
8. Derive the equation for conservation of mechanical energy for a freely falling body.
For a body of mass m dropped from height h:
- At the top: K.E. = 0, P.E. = mgh
- At ground: K.E. = ½mv2, P.E. = 0
- By conservation law: mgh = ½mv2
- Simplifies to: v = √(2gh)
9. What is the difference between energy conservation and energy conversion?
Energy conservation means the total energy in a closed system remains constant; it is not created or destroyed.
Energy conversion refers to the transformation from one form to another, like potential energy changing to kinetic energy or vice versa.
For example, as a ball falls, its potential energy is converted into kinetic energy, but the total mechanical energy remains conserved (if only conservative forces are present).
10. What are the conditions required for the law of conservation of mechanical energy to apply?
The law applies when:
- Only conservative forces (like gravity, spring force) act on the system
- No work is done by non-conservative forces (such as friction, air resistance)
- The system is isolated from external energy inputs or losses
11. Can you provide a solved numerical example on conservation of mechanical energy?
Example: A ball is dropped from a height of 20 m (g = 9.8 m/s2). Find its velocity just before hitting the ground.
Solution:
- P.E. at top = mgh
- K.E. at top = 0
- At bottom: all P.E. converts to K.E.
- mgh = ½mv2 ⇒ v = √(2gh)
- v = √(2 × 9.8 × 20) = √392 = 19.8 m/s
12. How can students practice and master problems on conservation of mechanical energy?
To master this topic:
- Solve a variety of numerical problems and worksheets
- Understand and apply formulas like K.E., P.E., and total mechanical energy
- Attempt conceptual MCQs and past exam questions
- Practice with topic-wise exercises available on Vedantu for stepwise solutions
