Know the principle of the conservation of energy and apply this principle to simple examples including the interpretation of simple flow diagrams

1.7.1 Energy

Learning Objective

Know the principle of the conservation of energy and apply this principle to simple examples, including the interpretation of simple flow diagrams.

Key Concepts

• Energy is the ability to do work or produce heat.
• Energy exists in many forms: kinetic, gravitational potential, elastic, chemical, thermal, electrical, etc.
• The total energy of an isolated system remains constant – the principle of conservation of energy.

Mathematical Formulation

The conservation of energy can be expressed as

\$\sum E{\text{initial}} = \sum E{\text{final}}\$

or, for a system where energy is transferred between forms,

\$\Delta E{\text{system}} = E{\text{in}} - E_{\text{out}}\$

where \$E{\text{in}}\$ and \$E{\text{out}}\$ are the rates of energy entering and leaving the system (in joules per second, i.e., watts).

Common Energy Forms and Their Symbols

Simple Flow Diagrams

A flow diagram shows how energy changes from one form to another. The arrows indicate the direction of transfer.

Example: A falling ball

\$Eg\$ (gravitational) → \$Ek\$ (kinetic)

Example: A stretched spring releasing a mass

\$Ee\$ (elastic) → \$Ek\$ (kinetic) → \$Eg\$ (gravitational) → \$E{th}\$ (thermal)

Applying the Principle – Worked Examples

1. Example 1 – Pendulum

   A 0.5 kg bob is released from a height of 0.20 m above its lowest point. Find its speed at the lowest point, ignoring air resistance.

   Solution:

   • Initial energy: \$E_{g}=mgh = 0.5 \times 9.8 \times 0.20 = 0.98\ \text{J}\$.
   • At the lowest point all this energy is kinetic: \$E_k = \frac{1}{2}mv^{2}\$.
   • Set \$E{g}=E{k}\$ → \$0.98 = \frac{1}{2}(0.5)v^{2}\$ → \$v^{2}=3.92\$ → \$v = 1.98\ \text{m s}^{-1}\$.

2. Example 2 – Spring‑loaded Toy Car

   A toy car of mass 0.2 kg is launched by a spring compressed 0.05 m. The spring constant is \$k = 800\ \text{N m}^{-1}\$. Find the speed of the car when the spring returns to its natural length, neglecting friction.

   Solution:

   • Elastic energy stored: \$E_e = \frac{1}{2}kx^{2}= \frac{1}{2}\times800\times(0.05)^{2}=1.0\ \text{J}\$.
   • All this becomes kinetic: \$E_k = \frac{1}{2}mv^{2}\$.
   • Set \$Ee = Ek\$ → \$1.0 = \frac{1}{2}(0.2)v^{2}\$ → \$v^{2}=10\$ → \$v = 3.16\ \text{m s}^{-1}\$.

3. Example 3 – Electrical Heater

   A 1500 W electric heater runs for 2 minutes. Calculate the thermal energy produced.

   Solution:

   • Power \$P = 1500\ \text{W} = 1500\ \text{J s}^{-1}\$.
   • Time \$t = 2\ \text{min} = 120\ \text{s}\$.
   • Thermal energy \$E_{th}=Pt = 1500 \times 120 = 1.8\times10^{5}\ \text{J}\$.

Interpreting Flow Diagrams – Practice Questions

1. In a simple electric circuit a battery supplies electrical energy to a resistor which heats up. Write the flow diagram and state the energy forms involved.
2. A roller coaster car at the top of a hill has 500 J of gravitational potential energy. After descending, it has 300 J of kinetic energy and the rest is lost as thermal energy due to friction. Represent this with a flow diagram.
3. Explain why the total energy in the system of Example 1 (pendulum) remains constant even though the forms of energy change.

Summary

• The total energy of an isolated system never changes – it is conserved.
• Energy can be transferred between different forms; flow diagrams help visualise these transfers.
• Quantitative problems are solved by equating the initial and final total energies, using the appropriate formulae for each form.