Introduction

Objects can store energy because of their motion, their position, or their deformation.

A moving bicycle has kinetic energy. A book on a shelf has gravitational potential energy. A compressed spring has elastic potential energy.

These forms are often grouped together as mechanical energy.

Mechanical energy can move between different stores, but the total amount of energy in a closed system remains constant. This is an application of the principle of conservation of energy.

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What is Mechanical Energy?

Mechanical energy is the energy associated with the motion and position of objects.

It includes:

• Kinetic energy
• Gravitational potential energy
• Elastic potential energy

The total mechanical energy of a system can therefore be written as:

$E_{\text{mech}}=E_k+E_g+E_e$Emech​=Ek​+Eg​+Ee​

where:

• $E_{\text{mech}}$Emech​ = total mechanical energy
• $E_k$Ek​ = kinetic energy
• $E_g$Eg​ = gravitational potential energy
• $E_e$Ee​ = elastic potential energy

Not every system contains all three stores.

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Kinetic Energy

The kinetic energy store is the energy an object has because it is moving.

Examples include:

• A rolling ball
• A moving car
• A running person
• A falling stone
• A swinging pendulum

Kinetic energy depends on:

• The object's mass
• The object's speed

A faster object has more kinetic energy.

A more massive object also has more kinetic energy when moving at the same speed.

The equation is:

$E_k=\frac12mv^2$Ek​=21​mv2

where:

• $m$m = mass in kilograms
• $v$v = speed in metres per second
• $E_k$Ek​ = kinetic energy in joules

Because speed is squared, doubling the speed gives four times as much kinetic energy.

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Gravitational Potential Energy

The gravitational potential energy store is energy associated with an object's position in a gravitational field.

An object gains gravitational potential energy when it is lifted.

Examples include:

• A book on a shelf
• Water behind a dam
• A roller-coaster car at the top of a hill
• A raised hammer
• A climber on a mountain

Near Earth's surface:

$E_g=mgh$Eg​=mgh

where:

• $m$m = mass
• $g$g = gravitational field strength
• $h$h = vertical height above a chosen reference level

The higher and more massive the object, the greater its gravitational potential energy.

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Elastic Potential Energy

The elastic potential energy store is energy stored when an elastic object is stretched or compressed.

Examples include:

• A stretched spring
• A compressed spring
• A drawn bow
• A stretched rubber band
• A compressed trampoline mat

For a spring obeying Hooke's Law:

$E_e=\frac12kx^2$Ee​=21​kx2

where:

• $k$k = spring constant
• $x$x = extension or compression
• $E_e$Ee​ = elastic potential energy

The more the spring is stretched or compressed, the more elastic energy it stores.

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Conservation of Energy

The principle of conservation of energy states:

Energy cannot be created or destroyed. It can only be transferred between stores.

In an ideal closed system:

$\text{Total energy before}=\text{Total energy after}$Total energy before=Total energy after

If there is no friction or air resistance, total mechanical energy remains constant.

This means:

$E_{k,i}+E_{g,i}+E_{e,i} = E_{k,f}+E_{g,f}+E_{e,f}$Ek,i​+Eg,i​+Ee,i​=Ek,f​+Eg,f​+Ee,f​

Energy can move from one mechanical store to another, but the total does not change.

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Mechanical Energy Transfers

Mechanical energy transformations occur when one store decreases while another increases.

For example:

• Falling object: gravitational potential → kinetic
• Rising object: kinetic → gravitational potential
• Released spring: elastic potential → kinetic
• Compressing a spring: kinetic or chemical → elastic potential
• Pendulum: gravitational potential ↔ kinetic

The transfer may be caused by forces doing work.

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Falling Objects

Consider a ball dropped from a height.

At the Top

• Gravitational potential energy is maximum.
• Kinetic energy is zero or very small.

While Falling

• Gravitational potential energy decreases.
• Kinetic energy increases.
• The ball speeds up.

Just Before Impact

• Gravitational potential energy is minimum.
• Kinetic energy is maximum.

Ignoring air resistance:

$\text{Loss in }E_g=\text{Gain in }E_k$Loss in Eg​=Gain in Ek​

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Rising Objects

Consider a ball thrown upward.

Just After Release

• Kinetic energy is large.
• Gravitational potential energy is relatively small.

While Rising

• Kinetic energy decreases.
• Gravitational potential energy increases.
• The ball slows down.

At the Highest Point

• Kinetic energy is momentarily zero.
• Gravitational potential energy is maximum.

Then the process reverses as the ball falls.

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Roller Coasters

A roller coaster is a classic example of mechanical energy transformation.

At the Top of the First Hill

• Gravitational potential energy is large.
• Kinetic energy is small.

Moving Downhill

• Gravitational potential energy decreases.
• Kinetic energy increases.
• The coaster speeds up.

At the Bottom

• Kinetic energy is large.
• Gravitational potential energy is small.

Moving Up the Next Hill

• Kinetic energy decreases.
• Gravitational potential energy increases.
• The coaster slows down.

In a real roller coaster, some mechanical energy is transferred into internal energy stores through friction and air resistance.

Therefore, without additional energy input, each later hill must be lower than the first.

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Pendulums

A pendulum continuously transfers energy between gravitational potential and kinetic stores.

At the Highest Point

• Gravitational potential energy is maximum.
• Kinetic energy is zero.
• Speed is zero.

Moving Downward

• Gravitational potential energy decreases.
• Kinetic energy increases.
• Speed increases.

At the Lowest Point

• Kinetic energy is maximum.
• Gravitational potential energy is minimum.
• Speed is greatest.

Moving Upward

• Kinetic energy decreases.
• Gravitational potential energy increases.
• Speed decreases.

In an ideal system, the pendulum would swing forever. In reality, air resistance and friction dissipate energy, so the pendulum gradually stops.

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Springs

A compressed or stretched spring stores elastic potential energy.

Compressed Spring at Rest

• Elastic potential energy is maximum.
• Kinetic energy is zero.

Released Spring

• Elastic potential energy decreases.
• Kinetic energy increases.
• The attached object speeds up.

Passing Through Equilibrium

• Kinetic energy is maximum.
• Elastic potential energy is minimum.

Stretching in the Opposite Direction

• Kinetic energy decreases.
• Elastic potential energy increases.

The energy repeatedly transfers between elastic potential and kinetic stores.

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Predicting Energy Changes

You can often predict energy changes without calculations.

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Mechanical Energy with Friction

When friction or drag acts, mechanical energy is not conserved by itself.

Some mechanical energy is transferred into:

• Internal energy stores
• Sound waves
• Deformation

Example: a sliding box

$\text{Kinetic store} \rightarrow \text{Mechanical work by friction} \rightarrow \text{Internal stores of box and floor}$Kinetic store→Mechanical work by friction→Internal stores of box and floor

Total energy is still conserved, but total mechanical energy decreases.

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Mechanical Energy vs Total Energy

This distinction is important.

Mechanical Energy

Includes:

• Kinetic
• Gravitational potential
• Elastic potential

Total Energy

Includes all energy stores, such as:

• Mechanical
• Internal
• Chemical
• Nuclear
• Electrostatic

In a system with friction:

• Mechanical energy may decrease.
• Total energy remains constant.

The missing mechanical energy has been transferred into other stores.

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Worked Example 1: Falling Object

A 2.0 kg object falls through 5.0 m.

Ignoring air resistance, the decrease in gravitational potential energy is:

$\Delta E_g=mgh$ΔEg​=mgh$=(2.0)(9.8)(5.0)$=(2.0)(9.8)(5.0)$=98\text{ J}$=98 J

Therefore, the kinetic energy increases by 98 J.

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Worked Example 2: Roller-Coaster Car

A roller-coaster car has 15,000 J of gravitational potential energy and 2,000 J of kinetic energy at one point.

Total mechanical energy:

$E_{\text{mech}}=15,000+2,000$Emech​=15,000+2,000$E_{\text{mech}}=17,000\text{ J}$Emech​=17,000 J

At a lower point, its gravitational potential energy is 5,000 J.

Ignoring friction:

$E_k=17,000-5,000$Ek​=17,000−5,000$E_k=12,000\text{ J}$Ek​=12,000 J

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Worked Example 3: Spring Launcher

A compressed spring stores 40 J of elastic potential energy.

When released, 32 J becomes kinetic energy.

The remaining energy transferred to other stores is:

$40-32=8\text{ J}$40−32=8 J

This 8 J may increase internal energy stores or be transferred by sound.

Total energy remains 40 J.

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Energy Bar Charts

An energy bar chart can show the relative amounts in each store.

For a falling object:

At the Top

• Gravitational: █████
• Kinetic:
• Internal:

Halfway Down

• Gravitational: ███
• Kinetic: ██
• Internal:

Near the Bottom

• Gravitational:
• Kinetic: █████
• Internal:

With air resistance, some bars would appear in the internal store.

Energy bar charts help students visualize conservation without using exact equations.

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Real-World Applications

Hydroelectric Power

Water stored behind a dam has gravitational potential energy.

As it falls:

$\text{Gravitational potential} \rightarrow \text{Kinetic} \rightarrow \text{Mechanical motion of turbine} \rightarrow \text{Electrical transfer}$Gravitational potential→Kinetic→Mechanical motion of turbine→Electrical transfer

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Bungee Jumping

As the jumper falls:

• Gravitational potential energy decreases.
• Kinetic energy increases.

As the cord stretches:

• Kinetic energy decreases.
• Elastic potential energy increases.

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Trampolines

As a person lands:

• Kinetic energy decreases.
• Elastic potential energy in the trampoline increases.

As the trampoline rebounds:

• Elastic potential energy decreases.
• Kinetic and gravitational potential energy increase.

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Vehicle Braking

When brakes are applied:

• Kinetic energy decreases.
• Internal energy stores of the brakes, tyres, and road increase.

The total energy is conserved, although mechanical energy decreases.

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Common Mistakes

Avoid these misconceptions:

• Mechanical energy is not only kinetic energy. It also includes gravitational and elastic potential energy.
• Energy is not used up. It is transferred between stores.
• An object at rest may still have mechanical energy if it has gravitational or elastic potential energy.
• Mechanical energy is not always conserved. Friction can transfer it into internal energy stores.
• Total energy is always conserved, even when mechanical energy decreases.
• A falling object loses gravitational potential energy, not total energy.

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Key Vocabulary

• Mechanical Energy – Energy associated with the motion and position of objects.
• Kinetic Energy – Energy stored by a moving object.
• Gravitational Potential Energy – Energy stored because of position in a gravitational field.
• Elastic Potential Energy – Energy stored in a stretched or compressed elastic object.
• Conservation of Energy – The principle that energy cannot be created or destroyed.
• Closed System – A system that does not exchange energy with its surroundings.
• Transformation – A change from one energy store to another.
• Dissipation – The spreading of energy into less useful stores, usually internal energy stores.
• Mechanical Work – Energy transferred when a force acts through a displacement.

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Summary

• Mechanical energy is the energy associated with the motion and position of objects.
• It includes kinetic, gravitational potential, and elastic potential energy.
• Mechanical energy can transfer between stores as objects move, rise, fall, speed up, slow down, or deform.
• Roller coasters, pendulums, springs, trampolines, and bungee jumps all demonstrate mechanical energy transformations.
• In an ideal closed system, total mechanical energy remains constant.
• When friction or drag acts, mechanical energy decreases because some energy is transferred into internal stores, but total energy is still conserved.
• Qualitative predictions can be made by identifying which stores increase and which decrease during motion.