Conservation of Mechanical Energy

Introduction

Imagine dropping a ball from a height. As it falls, its potential energy decreases while its kinetic energy increases. But the total energy (potential + kinetic) remains constant. This is the essence of the Conservation of Mechanical Energy.

Energy is one of the most fundamental concepts in physics. It exists in various forms such as mechanical, thermal, chemical, nuclear, and electrical. Among them, mechanical energy refers to the energy due to the motion and position of objects.

The principle of conservation of mechanical energy tells us that in the absence of non-conservative forces (like friction and air resistance), the total mechanical energy of a system remains constant.

This article explains the principle in detail, with mathematical proof, examples, and real-life applications.

What is Mechanical Energy?

Mechanical energy is the sum of Kinetic Energy (KE) and Potential Energy (PE) in a system. Emech=KE+PEE_{mech} = KE + PEEmech=KE+PE

Kinetic Energy (KE): Energy of motion.

KE=12mv2KE = \frac{1}{2}mv^2KE=21mv2

Potential Energy (PE): Energy of position or configuration. For gravitational potential energy,

PE=mghPE = mghPE=mgh

So, Emech=12mv2+mghE_{mech} = \frac{1}{2}mv^2 + mghEmech=21mv2+mgh

Law of Conservation of Mechanical Energy

Statement:

In an isolated system, where only conservative forces (like gravity or spring force) act, the total mechanical energy (KE + PE) remains constant.

KE+PE=constantKE + PE = \text{constant}KE+PE=constant

This principle is a special case of the Law of Conservation of Energy, which states that energy can neither be created nor destroyed, only transformed.

Conditions for Conservation

The system must be isolated (no external work).
Only conservative forces (gravity, spring force, electrostatic force) should act.
Non-conservative forces (friction, air resistance, viscosity) must be negligible.

Mathematical Proof (for Gravitational System)

Consider a body of mass mmm falling freely from height hhh.

At the top:
- Velocity = 0
- KE=0KE = 0KE=0
- PE=mghPE = mghPE=mgh
- Total Energy = mghmghmgh
At height yyy:
- Velocity = vvv
- By equation of motion:
v2=2g(h−y)v^2 = 2g(h-y)v2=2g(h−y)
- KE=12mv2=12m(2g(h−y))=mg(h−y)KE = \frac{1}{2}mv^2 = \frac{1}{2}m(2g(h-y)) = mg(h-y)KE=21mv2=21m(2g(h−y))=mg(h−y)
- PE=mgyPE = mgyPE=mgy
- Total Energy = mg(h−y)+mgy=mghmg(h-y) + mgy = mghmg(h−y)+mgy=mgh
At the bottom (ground):
- PE=0PE = 0PE=0
- KE=mghKE = mghKE=mgh
- Total Energy = mghmghmgh

👉 In all cases, Total Mechanical Energy remains the same.

Graphical Representation

PE vs Height: Straight line increasing with height.
KE vs Height: Decreasing line as height increases.
Total Energy vs Height: Constant horizontal line → Conservation.

Examples of Conservation of Mechanical Energy

1. Free Fall

Object falling under gravity.
PE decreases, KE increases.
Total remains constant.

2. Pendulum Motion

At extreme position: Maximum PE, Zero KE.
At mean position: Maximum KE, Minimum PE.
Total energy = constant.

3. Roller Coaster

At top of track: High PE, Low KE.
At bottom: High KE, Low PE.
Throughout motion: Total energy conserved.

4. Spring-Mass System

At stretched/compressed position: Maximum PE (elastic), zero KE.
At equilibrium: Maximum KE, zero PE.
Total = constant.

Conservation with Non-Conservative Forces

In real life, friction and air resistance are always present. They convert part of mechanical energy into heat, sound, or other forms.

Thus, KE+PE+Eloss=constantKE + PE + E_{loss} = \text{constant}KE+PE+Eloss=constant

Where ElossE_{loss}Eloss is energy lost to non-conservative forces.

Example: A ball bouncing loses height with each bounce because energy is lost as sound and heat.

Work-Energy Relation

From Work-Energy Theorem: W=ΔKEW = \Delta KEW=ΔKE

If only conservative forces act, the work done is stored as potential energy. Thus, ΔKE+ΔPE=0\Delta KE + \Delta PE = 0ΔKE+ΔPE=0 KE+PE=constantKE + PE = \text{constant}KE+PE=constant

This directly proves conservation of mechanical energy.

Numerical Examples

Example 1: Free Fall

A ball of mass 2 kg is dropped from 10 m. Find KE and PE at 6 m height.

Total Energy:

E=mgh=2×9.8×10=196JE = mgh = 2 \times 9.8 \times 10 = 196 JE=mgh=2×9.8×10=196J

At 6 m:

PE=mgh=2×9.8×6=117.6JPE = mgh = 2 \times 9.8 \times 6 = 117.6 JPE=mgh=2×9.8×6=117.6J KE=E−PE=196−117.6=78.4JKE = E – PE = 196 – 117.6 = 78.4 JKE=E−PE=196−117.6=78.4J

👉 KE = 78.4 J, PE = 117.6 J, Total = 196 J (constant).

Example 2: Pendulum

A pendulum bob of mass 1 kg swings to a height of 0.5 m. Find its velocity at the lowest point. PEtop=mgh=1×9.8×0.5=4.9JPE_{top} = mgh = 1 \times 9.8 \times 0.5 = 4.9 JPEtop=mgh=1×9.8×0.5=4.9J Atbottom,PE=0,KE=4.9JAt bottom, PE = 0, KE = 4.9 JAtbottom,PE=0,KE=4.9J 12mv2=4.9⇒v=2×4.91=3.13 m/s\frac{1}{2}mv^2 = 4.9 \quad \Rightarrow \quad v = \sqrt{\frac{2 \times 4.9}{1}} = 3.13 \, m/s21mv2=4.9⇒v=12×4.9=3.13m/s

👉 Velocity = 3.13 m/s.

Example 3: Spring System

A spring with k=100 N/mk = 100 \, N/mk=100N/m is compressed by 0.2 m. Find KE at equilibrium. PEspring=12kx2=12(100)(0.2)2=2JPE_{spring} = \frac{1}{2}kx^2 = \frac{1}{2}(100)(0.2)^2 = 2 JPEspring=21kx2=21(100)(0.2)2=2J

At equilibrium, all PE converts to KE = 2 J.

Applications in Real Life

Amusement Parks: Roller coasters designed using conservation of energy.
Engineering: Dams and hydroelectric plants rely on gravitational potential → kinetic → electricity.
Space Science: Satellites and planets conserve mechanical energy in orbits.
Sports: High jumps, pole vaults, swings rely on energy conversion.
Springs and Oscillations: Shock absorbers, clocks, trampolines.

Misconceptions About Conservation of Mechanical Energy

Misconception 1: Mechanical energy is always conserved.
- Wrong → Only conserved when non-conservative forces are absent.
Misconception 2: Kinetic and potential energy individually remain constant.
- Wrong → They constantly interchange, only total remains constant.
Misconception 3: Conservation of energy applies only to mechanics.
- Wrong → It’s a universal law across all branches of physics.

Advanced Concepts

1. Conservation in Planetary Motion

Planets orbiting the Sun continuously exchange KE and PE but total energy remains constant.

2. Conservation in Quantum Mechanics

Even at atomic level, electrons conserve mechanical energy when transitioning between states.

3. Energy Loss in Real Systems

Real systems include friction, drag, deformation → mechanical energy decreases, but total energy (including heat/sound) is still conserved.

Summary Table

Aspect	Conservation of Mechanical Energy
Definition	Total KE + PE of an isolated system remains constant
Formula	KE+PE=constantKE + PE = \text{constant}KE+PE=constant
Conditions	No non-conservative forces (friction, air drag)
Examples	Free fall, pendulum, roller coaster, spring system
Applications	Dams, satellites, roller coasters, oscillations
Graph	KE decreases while PE increases, total = constant