Physics (9702) | Energy conservation

Welcome to the Energy Conservation Chapter!

Ready to unlock one of the most fundamental laws in the entire universe? This chapter, Work, Energy, and Power, is central to Physics. It helps us understand why roller coasters stop, why engines heat up, and why energy bills exist! Don't worry if the formulas look challenging; we will break down the derivations step-by-step.

The central idea is simple: Energy is a shape-shifter. It can change from potential to kinetic, or electrical to heat, but the total amount stays exactly the same.

1. Understanding Work Done (W)

What is Work?

In Physics, the term Work Done has a very specific meaning, different from everyday usage (like doing homework or a job). Work is done only when a force causes an object to move some distance.

Definition: Work done is the product of the force and the displacement of the object in the direction of the force.

Key Formula:

$$W = Fs$$
Where:
\(W\) is Work Done (in Joules, J)
\(F\) is the Force (in Newtons, N)
\(s\) is the distance moved in the direction of the force (in metres, m)

The Direction Rule (Crucial Point!)

Work is a scalar quantity, but its calculation depends heavily on the direction of the force and displacement.

• If you lift a box vertically, the work done on the box is \(W = F \times h\), where \(F\) is the weight of the box and \(h\) is the height.
• If you carry that same box horizontally across the room, the vertical force (weight) does zero work because the displacement is perpendicular to the force.

Analogy: The Tug-of-War Component

Imagine pulling a heavy sled using a rope at an angle. Your effort is split: some effort pulls up, and some pulls forward. Only the component of the force that is parallel to the direction of motion actually does work.

If the force and displacement are perpendicular (90°), no work is done!

Key Takeaway: Work is energy transferred by a force. If motion occurs against the force (like friction), that force does negative work (energy is removed from the system).

2. The Law of Energy Conservation

The Principle of Conservation of Energy (PoCE)

This is the big one! Everything in this chapter stems from this law.

State and Apply: The Principle of Conservation of Energy states that energy cannot be created or destroyed, but only transformed from one form to another.

In any closed system (a system where no energy enters or leaves), the total energy remains constant:

$$E_{\text{Total Initial}} = E_{\text{Total Final}}$$

Analogy: Roller Coasters

A perfect example is a roller coaster car on a track (ignoring air resistance and friction, for a moment). As the car climbs to the highest point, it gains maximum Gravitational Potential Energy (GPE). As it plunges, this GPE transforms entirely into Kinetic Energy (KE).

• Top of the hill: \(E_{\text{Total}} = E_p (\text{max}) + E_k (0)\)
• Bottom of the hill: \(E_{\text{Total}} = E_p (0) + E_k (\text{max})\)

The sum of GPE and KE is conserved throughout the ride.

Key Takeaway: When solving conservation problems, identify all initial energy forms and equate them to all final energy forms (plus any work done against resistive forces).

3. Gravitational Potential Energy (\(\Delta E_p\)) and Kinetic Energy (\(E_k\))

3.1 Gravitational Potential Energy Change (\(\Delta E_p\))

This is the energy stored in an object due to its height in a uniform gravitational field (like the surface of the Earth).

Recall and Use Formula:

$$\Delta E_p = mg\Delta h$$

Where:
\(\Delta E_p\) is the change in GPE (J)
\(m\) is mass (kg)
\(g\) is acceleration of free fall (or gravitational field strength, \(\text{N/kg}\) or \(\text{m/s}^2\))
\(\Delta h\) is the change in height (m)

Step-by-Step Derivation of \(\Delta E_p = mg\Delta h\) (Syllabus 5.2.1)

1. We start with the definition of work done: $$W = Fs$$
2. For an object being lifted, the force required (\(F\)) must be equal to the object's weight. Recall that Weight is \(W = mg\). So, \(F = mg\).
3. The distance moved (\(s\)) is the change in height (\(\Delta h\)).
4. Substitute these into the work equation: $$W = (mg)(\Delta h)$$
5. Since the work done in lifting the object is stored as gravitational potential energy, $$\Delta E_p = mg\Delta h$$

This shows that the potential energy gained is equal to the work done against gravity.

3.2 Kinetic Energy (\(E_k\))

This is the energy an object possesses due to its motion.

Recall and Use Formula:

$$E_k = \frac{1}{2}mv^2$$

Where:
\(E_k\) is Kinetic Energy (J)
\(m\) is mass (kg)
\(v\) is speed (m/s)

Step-by-Step Derivation of \(E_k = \frac{1}{2}mv^2\) (Syllabus 5.2.3)

We derive this using the work done ($W$) to accelerate an object from rest ($u=0$) to a speed ($v$) over a displacement ($s$) under a constant force ($F$).

1. Start with the definition of work done: $$W = Fs$$
2. Use Newton's Second Law: $$F = ma$$
3. Substitute $F$ into the work equation: $$W = (ma)s$$
4. Use the kinematic equation (SUVAT) that does not involve time: $$v^2 = u^2 + 2as$$
5. Since the object starts from rest, $u=0$: $$v^2 = 2as$$
6. Rearrange this to find the displacement $s$: $$s = \frac{v^2}{2a}$$
7. Substitute this expression for $s$ back into the work equation from step 3: $$W = ma \left( \frac{v^2}{2a} \right)$$
8. Cancel out the acceleration $a$: $$W = \frac{1}{2}mv^2$$
9. Since the work done in accelerating the object is stored as kinetic energy: $$E_k = \frac{1}{2}mv^2$$

Key Takeaway: Mechanical energy transformations (like a ball bouncing) involve the trade-off between \(E_p\) (height) and \(E_k\) (speed).

4. Efficiency

No real-world machine is perfect! When energy is transformed, some of it is always 'wasted,' usually as thermal energy (heat) or sound.

Definition: Efficiency is the ratio of useful energy (or power) output from a system to the total energy (or power) input to the system.

Efficiency Formulae (Syllabus 5.1.3 & 5.1.4):

In terms of energy:

$$\text{Efficiency} = \frac{\text{Useful Energy Output}}{\text{Total Energy Input}} \times 100\%$$

In terms of power:

$$\text{Efficiency} = \frac{\text{Useful Power Output}}{\text{Total Power Input}} \times 100\%$$

• Efficiency is usually expressed as a percentage, but can also be a decimal value (e.g., 0.8 is 80% efficient).
• Efficiency is a unitless quantity because it is a ratio of two identical quantities (J/J or W/W).
• By the PoCE, the total energy input must equal the useful energy output plus the wasted energy. This means efficiency can never be greater than 100%.

Did You Know?

The energy conservation law dictates that the energy doesn't disappear; it just becomes 'useless' for the machine's intended task. For instance, the heat generated by a laptop is conserved energy, but it's wasted energy because the laptop's purpose is not to act as a heater!

Key Takeaway: Efficiency tells you how good a device is at converting input energy into the desired output energy. Real systems always waste energy.

5. Power (P)

Definition of Power

Energy transfer takes time. Power measures how quickly that transfer happens.

Definition: Power is defined as the work done per unit time, or the rate of energy transfer (Syllabus 5.1.5).

Recall and Use Formula:

$$P = \frac{W}{t}$$
Where:
\(P\) is Power (in Watts, W)
\(W\) is Work Done (in Joules, J)
\(t\) is Time taken (in seconds, s)

The unit Watt (W) is equivalent to one Joule per second ($1 \text{ W} = 1 \text{ J/s}$).

Derivation of Power in terms of Force and Velocity ($P=Fv$)

This is an extremely useful formula, especially when dealing with vehicles moving at a constant speed against resistive forces (like air resistance and friction).

Step-by-Step Derivation of $P = Fv$ (Syllabus 5.1.7)

1. Start with the definition of power: $$P = \frac{W}{t}$$
2. Substitute the formula for work done (\(W = Fs\)): $$P = \frac{Fs}{t}$$
3. Recall the definition of velocity (or speed) as displacement over time: $$v = \frac{s}{t}$$
4. Substitute $v$ into the power equation: $$P = Fv$$

Application of $P=Fv$

When a car moves at a constant speed, the engine must produce a driving force $F$ that exactly balances the total resistive forces (air resistance, friction, etc.).

If the engine operates at constant power $P$, then:

$$v = \frac{P}{F}$$

This shows an inverse relationship: $v \propto 1/F$. If the car goes faster, the driving force it can maintain decreases. This is why it's hard to accelerate once you are already going very fast—the engine power is being stretched to overcome the huge air resistance.

Key Takeaway: Power is the speed limit of energy transfer. The relation $P=Fv$ allows you to find the instantaneous power needed to produce a force at a specific velocity.

Chapter Summary & Final Review

Syllabus Check-In: Essential Formulas

Make sure you can recall, use, and derive these three core equations:

1. Work Done: \(W = Fs\) (where $s$ is displacement in the direction of $F$)
2. Gravitational Potential Energy Change: $$\Delta E_p = mg\Delta h$$
3. Kinetic Energy: $$E_k = \frac{1}{2}mv^2$$

And remember the key concepts:

• The Principle of Conservation of Energy is the foundation for all energy calculations. Energy is transformed, never lost.
• Efficiency is always less than 100% in real systems due to wasted energy (e.g., heat).
• Power is the rate of energy transfer, calculated as $P = W/t$ or, critically for motion problems, $P = Fv$.

You've got this! Practice mixing these concepts in problems—that's where conservation shines!