Physics (9630) | Conservation of energy

📚 Conservation of Energy: Your Essential Study Guide (Physics 9630) 🚀

Hello future physicists! This chapter is one of the most fundamental and important concepts in all of science. It’s all about tracking energy—how it moves, how it changes form, but crucially, how the total amount always stays the same.
Mastering this topic is essential because energy transformations underpin everything in mechanics, from a swinging pendulum to a car braking. Don't worry if the calculations look complex; we will break down the rules and the formulas step-by-step!

3.2.7 Work, Energy, and Power

Understanding Work Done (W)

In physics, "work" has a very specific definition. Work is done when a force causes an object to move in the direction of that force. It is the measure of energy transferred.

The Work Done Formula:

\(W = Fs \cos \theta\)

• \(W\): Work done (measured in Joules, J).
• \(F\): The constant force applied (Newtons, N).
• \(s\): The displacement (distance moved) in metres (m).
• \(\theta\): The angle between the direction of the force (\(F\)) and the direction of displacement (\(s\)).

💡 Why is \(\cos \theta\) important?
The \(\cos \theta\) component ensures you only count the part of the force that is *parallel* to the movement.

• If you push a block straight along the floor, force and displacement are parallel, so \(\theta = 0^{\circ}\). Since \(\cos(0^{\circ}) = 1\), \(W = Fs\). This is maximum work.
• If you carry a heavy suitcase horizontally, the force (gravity/weight) is vertical, but the displacement is horizontal. \(\theta = 90^{\circ}\). Since \(\cos(90^{\circ}) = 0\), the work done by gravity is zero! (You are doing work to hold it up, but zero work is done by the vertical force of gravity over the horizontal displacement.)

Key Takeaway: Graphical Representation
The amount of work done is equal to the area under a force-displacement graph. This is especially useful if the force applied is not constant.

Power (P): The Rate of Doing Work

Power tells you how quickly work is being done or how quickly energy is being transferred.

Definition: Power is the rate of doing work (or the rate of energy transfer).

\(P = \frac{\Delta W}{\Delta t}\) (or \(P = \frac{\Delta E}{\Delta t}\))

• \(P\): Power (measured in Watts, W, where 1 W = 1 J s\(^{-1}\)).
• \(\Delta W\): Change in work done (J).
• \(\Delta t\): Change in time (s).

A Useful Derived Formula (For Constant Velocity):
You can also express power in terms of force and velocity.

\(P = Fv\)

This formula is super handy for calculating the power needed to keep a car moving at a constant velocity (\(v\)) against a constant resistive force (\(F\)).

Efficiency

No machine is 100% efficient! Efficiency is a measure of how much of the energy put into a system is converted into the useful output energy (rather than wasted energy, usually heat).

\(\text{Efficiency} = \frac{\text{useful output power}}{\text{input power}}\)

Efficiency is often expressed as a percentage by multiplying the result by 100.

3.2.8 The Principle of Conservation of Energy

The Golden Rule

This is the core concept of the chapter.

The Principle of Conservation of Energy (PCE) states that energy cannot be created or destroyed; it can only be transferred from one form to another, or from one body to another.
Therefore, the total energy in a closed system remains constant.

Analogy: Think of energy as a fixed amount of cash you have. You can move it from your Savings Account (GPE) to your Checking Account (KE) or use it to pay a bill (Work Done), but the total amount of cash you possess never changes.

Forms of Mechanical Energy

In mechanics problems, we primarily deal with three types of energy that are interconvertible:

1. Kinetic Energy (\(E_k\))

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

\(E_k = \frac{1}{2}mv^2\)

• \(m\): mass (kg)
• \(v\): velocity (m s\(^{-1}\))

2. Gravitational Potential Energy (\(E_p\))

This is the energy an object possesses due to its position in a gravitational field, usually measured relative to a zero reference point (like the ground).

\(\Delta E_p = mgh\)

• \(m\): mass (kg)
• \(g\): acceleration due to gravity (N kg\(^{-1}\) or m s\(^{-2}\))
• \(h\): change in vertical height (m)

3. Elastic Potential Energy (\(E_{el}\)) (Often covered in Section 3.2.9 - Bulk Properties, but essential for conservation problems involving springs or stretching)

This is the energy stored in an object (like a spring or a stretched wire) when work is done to deform it elastically.

The energy stored is equal to the area under the force-extension graph.

\(E_{el} = \frac{1}{2} F \Delta L\)

(Note: \(\Delta L\) represents the extension or compression of the spring/wire.)

Applying Conservation of Energy (The Formula Setup)

In a typical physics problem, energy is conserved between two points (A and B).

The general setup is:

\(\text{Total Energy at A} = \text{Total Energy at B}\)

If we consider only KE and GPE and assume no friction or air resistance (ideal case):

\(E_{k, A} + E_{p, A} = E_{k, B} + E_{p, B}\)

Step-by-Step for Real-World Problems (Including Resistance)

The syllabus requires quantitative and qualitative application involving GPE, KE, EPE, and work done against resistive forces.

When non-conservative forces like drag (air resistance) or friction are present, they do work which transfers mechanical energy (KE/GPE) into heat (thermal energy). This energy is "lost" from the mechanical system.

The conservation equation becomes:

\(E_{\text{Input}} = E_{\text{Output (Useful)}} + E_{\text{Wasted (Work against resistance)}}\)

In a falling object problem (A to B):

\((\text{KE}_A + \text{GPE}_A) = (\text{KE}_B + \text{GPE}_B) + \text{Work done against resistance (W}_{R})\)

⚠️ Common Mistake Alert: Remember that \(W_R\) is energy lost. If a question asks for the minimum energy required to lift an object, you must calculate the gain in GPE + the work done against drag/friction.

3.2.9 Energy and Elasticity (Revisited)

As covered briefly above, understanding how energy is stored in deformed materials is crucial for conservation problems involving springs or stretching.

Energy Stored in a Spring or Wire

When you stretch or compress a spring or wire within its elastic limit (following Hooke's Law, \(F = k\Delta L\)), the work done is stored as Elastic Potential Energy (\(E_{el}\)).

Since the force increases linearly with extension (\(\Delta L\)), the work done is calculated from the area of the force-extension triangle:

\(\text{Energy stored} = \frac{1}{2} \times \text{Base} \times \text{Height}\)

\(E_{el} = \frac{1}{2} F \Delta L\)

If you substitute Hooke's Law (\(F = k \Delta L\)) into this equation, you get the alternative form (which may be useful in certain contexts):

\(E_{el} = \frac{1}{2} k (\Delta L)^2\)

Energy Transformations involving Elasticity

Energy conservation principles apply perfectly to elastic systems:

• Example 1: A mass fired vertically by a spring.
  When the spring is compressed, it stores EPE. When released, this EPE transforms into KE. As the mass flies upward, the KE transforms into GPE.

\(\text{Initial EPE} \rightarrow \text{Maximum KE} \rightarrow \text{Maximum GPE}\)

If we ignore air resistance: \(\frac{1}{2} k (\Delta L)^2 = \frac{1}{2} mv^2 = mgh_{\text{max}}\)

Key Takeaway: Energy and Deformation
Energy stored elastically is recoverable (like a rubber band snapping back). Energy used in plastic deformation (permanent stretch) is typically converted to heat and is not recovered, illustrating the energy required to permanently deform a solid.