4.1.1.1 Energy Stores and Systems

In Physics, energy is a property that must be transferred to an object in order to perform work on, or to heat, that object. It is never "used up"; it is only moved from one store to another.

1. What is a System?

In the context of GCSE Physics, a system is simply an object or a group of objects that you are choosing to study.

1. Closed System: A system where no energy can enter or leave. The total energy in a closed system always remains constant, though it can be transferred between different stores within the system.
2. Open System: Energy can be exchanged with the surroundings (e.g., heat escaping into the air).

2. The 8 Energy Stores

You need to be able to identify the store in which energy is held. Use the acronym KG CEMENT to remember them:

3. Energy Transfers (Pathways)

Energy moves between stores via four main pathways:

1. Mechanical Work: A force moving an object through a distance.
2. Electrical Work: Charges moving due to a potential difference.
3. Heating: Due to temperature differences.
4. Radiation: Energy transferred by waves (e.g., light or sound).

4. Mathematical Representations of Energy

Kinetic Energy: The energy of a moving object depends on its mass and its speed.

$E_k$ = $\frac{1}{2}m{v}^2$

1. $E_k$: Kinetic energy in Joules (J)
2. m: Mass in kilograms (kg)
3. v: Speed in metres per second (m/s)

Gravitational Potential Energy ($E_p$): The energy gained by an object raised above the ground.

$E_p$ = mgh

1. $E_p$: Gravitational potential energy in Joules (J)
2. m: Mass in kilograms (kg)
3. g: Gravitational field strength (9.8 N/kg on Earth)
4. h: Height in metres (m)

Elastic Potential Energy ($E_e$): The energy stored in a stretched spring.

$E_e=\frac{1}{2}k{e}^2$

1. $E_e$: Elastic potential energy in Joules (J)
2. k: Spring constant in Newtons per metre (N/m)
3. e: Extension in metres (m)

5. Edge Cases and Common Misconceptions

The Speed Squared Pitfall: In the kinetic energy formula, only the velocity is squared ($v^2$). If an object's speed doubles, its kinetic energy quadruples because $2^2$ = 4.

Weight vs. Mass: If an exam question gives you the weight in Newtons, you do not need to multiply by g again in the $E_p$ formula, as Weight = $m\times g$.

Worked Example: The Falling Ball

A $2\text{ kg}$ ball is held $10\text{ m}$ above the ground. It is released and falls. Calculate the kinetic energy of the ball just before it hits the ground (ignore air resistance).

Step 1: Calculate initial $E_p$

$E_p$ = mgh = $2\times 9.8\times 10$ = $196\text{ J}$

Step 2: Apply Conservation of Energy

In a closed system, the energy lost from the gravitational store is equal to the energy gained in the kinetic store.

$E_p\text{ lost}$ = $E_k\text{ gained}$

Therefore, the $E_k$ just before impact is $196\text{ J}$.

4.1.1.2 Changes in Energy

When a system changes, energy is transferred. To excel in your GCSEs, you must be able to calculate the exact amount of energy held in the three primary "mechanical" stores: Kinetic, Gravitational Potential, and Elastic Potential.

1. Kinetic Energy ($E_k$)

Anything that moves has energy in its kinetic store. The amount of energy depends on the object's mass and the square of its speed.

The Formula:

$E_k$ = $\frac{1}{2}m{v}^2$

1. $E_k$ = Kinetic energy in Joules (J)
2. m = Mass in kilograms (kg)
3. v = Speed in metres per second (m/s)

Expert Tip: In the AQA exam, they often give you the mass in grams. Always convert to kg (divide by 1,000) before plugging it into the formula!

2. Gravitational Potential Energy ($E_p$)

When an object is raised in a gravitational field, it requires work to be done against the force of gravity. This energy is stored as $E_p$.

The Formula:

$E_p$ = mgh

1. $E_p$ = Gravitational potential energy in Joules (J)
2. m = Mass in kilograms (kg)
3. g = Gravitational field strength in Newtons per kilogram (N/kg) (On Earth, this is $9.8\text{ N/kg}$)
4. h = Height in metres (m)

3. Elastic Potential Energy ($E_e$)

When you stretch or compress an object (like a spring or a rubber band), energy is stored in its elastic store. This is true as long as the object has not been stretched past its limit of proportionality.

The Formula:

$E_e$ = $\frac{1}{2}k{e}^2$

1. $E_e$ = Elastic potential energy in Joules (J)
2. k = Spring constant in Newtons per metre (N/m)
3. e = Extension (or compression) in metres (m)

4. Mathematical Example: The Archery Challenge

An archer pulls back a bowstring with a spring constant of $200\text{ N/m}$. The string is pulled back by $0.4\text{ m}$.

1. Calculate the energy stored in the bow.
2. If the arrow has a mass of $0.05\text{ kg}$, calculate the maximum speed of the arrow when released (assume 100% efficiency).

Part 1 Solution:

$E_e$ = $\frac{1}{2}k{e}^2$

$E_e$ = $0.5\times 200\times (0.4{)}^2$

$E_e$ = $100\times 0.16=16\text{ J}$

Part 2 Solution:

Assume all $E_e$ transfers to $E_k$:

16 = $\frac{1}{2}m{v}^2$

16 = $0.5\times 0.05\times v^2$

16 = $0.025\times v^2$

$v^2=\frac{16}{0.025}=640$

$v=\sqrt{640}\approx 25.3\text{ m/s}$

5. Edge Cases & Comparison

1. The "Limit of Proportionality": If a spring is stretched too far, the $E_e$ formula no longer works because the relationship between force and extension is no longer linear.
2. Energy Conservation: In many exam questions, you will need to set $E_p=E_k$ (for a falling object) or $E_e=E_k$ (for a catapult/spring).
3. Units Matter: If the extension (e) is given in cm, you must convert it to m (divide by 100) before squaring it.

4.1.1.3 Energy Changes in Systems: Specific Heat Capacity

1. Definition: The Specific Heat Capacity of a substance is the amount of energy required to raise the temperature of 1 kg of the substance by 1^\circ\text{C}.
2. Explanation: Different materials require different amounts of energy to heat up. Water has a very high specific heat capacity, meaning it takes a lot of energy to increase its temperature.
3. Formula: $\Delta E$ = $mc\Delta \theta$
4. Units: Change in Thermal Energy ($\Delta E$): Joules (J)
5. Mass (m): Kilograms (kg)
6. Specific Heat Capacity (c): Joules per kilogram per degree Celsius ($J/k{g}^{\circ }\text{C}$)
7. Temperature Change ($\Delta \theta$): Degrees Celsius (${}^{\circ }\text{C}$)

4.1.1.4 Power

1. Definition: Power is defined as the rate at which energy is transferred or the rate at which work is done.
2. Explanation: An object with high power performs the same amount of work in a shorter time than one with low power.
3. Formulas: $P=\frac{E}{t}$ or $P=\frac{W}{t}$
4. Units: * Power (P): Watts (W) — Note: 1 Watt = 1 Joule per second.
5. Energy Transferred (E) or Work Done (W): Joules (J)
6. Time (t): Seconds (s)

Math Example:

Two motors lift the same 50N weight 2 metres high. Motor A takes 5s, Motor B takes 10s.

1. Work Done (W = F * d): 50 * 2 = 100 J
2. Power A: 100 / 5 = 20 W
3. Power B: 100 / 10 = 10 W
4. Conclusion: Motor A is more powerful because it transfers the same energy faster.