Theory Exercises

Mechanical Energy

Definition

Mechanical energy is the sum of kinetic energy and potential energy in a system.

\[E_m = KE + PE = \frac{1}{2}mv^2 + mgh\]

Where:
  • KE = kinetic energy (energy of motion)
  • PE = potential energy (stored energy due to position)

Types of Mechanical Energy

1. Kinetic Energy (KE)

Energy of motion:

\[KE = \frac{1}{2}mv^2\]

Characteristics:
  • Zero when object is at rest
  • Increases with velocity squared
  • Always positive

2. Gravitational Potential Energy (PE)

Energy stored due to height:

\[PE = mgh\]

Characteristics:
  • Relative to chosen reference level
  • Higher position = more PE
  • Zero at reference level

3. Elastic Potential Energy

Energy stored in compressed/stretched materials:

\[PE_e = \frac{1}{2}kx^2\]

Conservation of Mechanical Energy

The Principle

In the absence of friction and other dissipative forces, the total mechanical energy of a system remains constant.

\[E_{m,\text{total}} = KE + PE = \text{constant}\]

Mathematical Expression

At any two points in motion:

\[KE_1 + PE_1 = KE_2 + PE_2\]

\[\frac{1}{2}mv_1^2 + mgh_1 = \frac{1}{2}mv_2^2 + mgh_2\]

Example: Falling Ball

A 2 kg ball dropped from 20 m height (g = 10 m/s²):

Initial state (at rest, 20 m high):
  • KE₀ = 0 J
  • PE₀ = 2 × 10 × 20 = 400 J
  • E_m = 400 J
At 10 m height:
  • PE = 2 × 10 × 10 = 200 J
  • KE = 400 - 200 = 200 J
  • E_m = 400 J
At ground level:
  • PE = 0 J
  • KE = 400 J
  • E_m = 400 J

Notice: Total energy stays constant at 400 J throughout!

Energy Transformation Scenarios

1. Pendulum Swing

A pendulum demonstrates continuous KE ↔ PE transformation:

At highest point (extreme):
  • KE = minimum (zero at extreme point)
  • PE = maximum
  • Total = constant
At lowest point (bottom):
  • KE = maximum
  • PE = minimum
  • Total = constant
The transformation continues indefinitely without friction.

2. Roller Coaster (Frictionless)

Energy converts as cart moves:

At hill top:
  • High PE, low KE
  • Total mechanical energy constant
Going down:
  • PE decreases, KE increases
  • At bottom: maximum KE, minimum PE
Going back up:
  • KE decreases, PE increases
  • Cart can reach same height as start

3. Launched Projectile

An object thrown upward at angle:

At launch:
  • KE = maximum (full speed)
  • PE = 0 (at reference)
  • E_m = total
At peak height:
  • KE = minimum (only horizontal component)
  • PE = maximum
  • E_m = same total
At landing:
  • PE = 0 (back at reference)
  • KE = same as launch (speed recovered)
  • E_m = same total

Mechanical Energy with Friction

Reality: Friction Always Present

In real-world scenarios, friction and air resistance dissipate energy:

\[E_{m,\text{initial}} = E_{m,\text{final}} + E_{\text{friction}}\]

The "lost" mechanical energy converts to:

  • Thermal energy (heat)
  • Sound energy
  • Deformation energy

Calculating Energy Lost to Friction

Work done by friction:
\[W_f = f \times d = \mu N \times d\]
Energy dissipated:
\[E_{\text{lost}} = W_f\]

Example: Sliding Block

A 5 kg block slides 10 m on a surface with μ_k = 0.2:

Friction force:
\[f = \mu N = 0.2 × (5 × 10) = 10 \text{ N}\]
Work done by friction:
\[W_f = 10 × 10 = 100 \text{ J}\]
Energy converted to heat:
\[E_{\text{heat}} = 100 \text{ J}\]

This energy heats the surfaces in contact.

Mechanical Advantage and Energy

Machines and Efficiency

Machines transfer mechanical energy, but some is lost to friction:

\[\text{Efficiency} = \frac{\text{Useful Energy Out}}{\text{Total Energy In}} × 100\%\]

Simple Machine Examples

Inclined Plane

  • Spreads work over distance
  • Less force needed but longer distance
  • Energy input = work output + friction losses

Pulley System

  • Changes force direction
  • Reduces force but increases distance
  • Energy efficiency depends on friction

Lever

  • Multiplies force at expense of distance
  • Work = Force × Distance (unchanged by lever)
  • Energy conserved (ignoring friction)

Calculating Mechanical Energy

Step-by-Step Approach

  1. Identify initial and final states
  2. Choose reference level for PE (usually ground)
  3. Calculate KE₁ = 0.5 × m × v₁²
  4. Calculate PE₁ = m × g × h₁
  5. Find total: E_m = KE₁ + PE₁
  6. Repeat for final state
  7. Check conservation: E_m,initial = E_m,final + E_friction

Example Problem

Scenario: A 3 kg ball is thrown upward at 10 m/s from ground level.
  • What is total mechanical energy?
  • What is maximum height reached?
  • What is KE at 2 m height?
Solution: Initial (at ground, v = 10 m/s):
  • KE₀ = 0.5 × 3 × (10)² = 150 J
  • PE₀ = 0 J
  • E_m = 150 J
At maximum height (v = 0):
  • KE = 0 J
  • PE = mgh = E_m = 150 J
  • h = 150 / (3 × 10) = 5 m
At h = 2 m:
  • PE = 3 × 10 × 2 = 60 J
  • KE = 150 - 60 = 90 J

Real-World Applications

Energy-Generating Systems

Hydroelectric Power:
  • Water PE converts to KE
  • KE drives turbine generator
  • Converts mechanical to electrical energy
Wind Energy:
  • Air KE in wind
  • Turns turbine blades
  • Generator converts to electrical energy
Geothermal Power:
  • Internal PE of hot rocks
  • Heat drives system
  • Thermal to mechanical to electrical

Sport and Recreation

Skateboard Ramp:
  • Gravitational PE converts to KE going down
  • KE converts back to PE going up other side
  • Friction removes energy gradually
Springboard Diving:
  • Diver's PE high on board
  • Board's elastic PE stores energy
  • Releases as KE launching diver
  • PE increases as diver rises and falls
Trampoline:
  • Elastic PE in trampoline
  • Converts to jumper's KE
  • KE converts to PE at height
  • Returns to elastic PE on landing

Key Takeaways

  1. Mechanical energy = KE + PE = constant (without friction)
  2. Energy transformation occurs continuously in systems
  3. At highest point: Maximum PE, minimum KE
  4. At lowest point: Minimum PE, maximum KE
  5. Conservation law: E_initial = E_final + E_friction_losses
  6. Friction dissipates energy as heat and sound
  7. Machines transfer mechanical energy with losses due to friction