Physics (9702) | Work, energy and power

AS Level Physics (9702) Study Notes: Work, Energy and Power

Welcome to one of the most fundamental chapters in Physics! Work, Energy, and Power describe how things move, how forces affect them, and how quickly these processes happen. This topic is essential because it links mechanics (forces and motion) to the concepts of efficiency and consumption we see every day. Understanding these principles will help you tackle complex problems in Kinematics and Dynamics later on. Let's get started!

Don't worry if the derivations seem long; focusing on the core definitions and the final formulas is the most important step for success!

5.1 Defining Work Done (W)

In physics, "work" has a very specific meaning. It isn't just about feeling tired after studying!

Definition of Work Done

Work Done (W) is the energy transferred when a force causes an object to move over a distance. It is a scalar quantity (it only has magnitude, not direction).

Key Formula:
Work Done = Force $\times$ Displacement in the direction of the force
$$W = Fs$$

Where:

• \(W\) is the work done (measured in Joules, J).
• \(F\) is the magnitude of the force (measured in Newtons, N).
• \(s\) is the displacement (measured in metres, m).

Units Check: Homogeneity

Since \(W = Fs\), the unit of Work is $N \cdot m$. We define one Joule (J) as the work done when a force of 1 Newton moves an object 1 metre in the direction of the force.
$$1 \text{ J} = 1 \text{ N m}$$

The Critical Angle Condition

What if the force is applied at an angle to the direction of motion? Only the component of the force acting in the direction of the displacement does work.

If the force $F$ acts at an angle $\theta$ to the displacement $s$, the useful component of the force is $F \cos\theta$.
$$W = (F \cos\theta)s$$

Analogy: Imagine pulling a sled across the snow. You pull the rope upwards at an angle ($\theta$). Only the forward-pulling part of your force ($F \cos\theta$) actually moves the sled forward and does useful work. The vertical part ($F \sin\theta$) does no work because the sled is not displaced vertically.

Quick Review: When is Work Done Zero?
Work done is zero if:
1. The displacement \(s\) is zero (you push a wall, it doesn't move).
2. The force \(F\) is perpendicular to the displacement ($\theta = 90^\circ$). (e.g., a satellite orbiting Earth: gravity pulls inward, but the movement is tangential. Gravity does no work on the satellite).

5.1 Energy Conservation (The Big Picture)

Work done is just a measure of energy transfer. This leads us to the most important principle in physics.

Principle of Conservation of Energy (PCE)

The Principle of Conservation of Energy states that energy cannot be created or destroyed, but it can be changed from one form to another. The total energy in a closed system remains constant.

Example: When you drop a ball, its total energy remains the same. Potential energy transforms into kinetic energy, and some energy is lost as thermal energy (heat) and sound energy due to air resistance.

In problem-solving using PCE, we often state:
$$\text{Loss of one form of energy} = \text{Gain of another form of energy}$$

5.2 Gravitational Potential Energy ($E_p$)

Definition and Derivation of $\Delta E_p = mg\Delta h$

Gravitational Potential Energy ($\mathbf{E_p}$) is the energy stored in an object due to its position within a gravitational field (i.e., its height).

We are asked to derive the formula for the change in $E_p$ ($\Delta E_p$) in a uniform gravitational field (like near the Earth’s surface) using $W = Fs$.

Step-by-step Derivation:
1. To lift an object of mass $m$ at a constant velocity, the upward lifting force $F$ must be equal to its weight $W_g$.
$$F = \text{Weight} = mg$$
2. The displacement $s$ is the change in height, $\Delta h$.
$$s = \Delta h$$
3. The work done to lift the object is $W = Fs$. This work done is stored as Gravitational Potential Energy ($\Delta E_p$).
$$\Delta E_p = W = (mg)(\Delta h)$$
Final Formula:
$$\Delta E_p = mg\Delta h$$

5.2 Kinetic Energy ($E_k$)

Definition and Derivation of $E_k = \frac{1}{2}mv^2$

Kinetic Energy ($\mathbf{E_k}$) is the energy possessed by an object due to its motion. It depends on both mass and speed.

We are asked to derive the formula for $E_k$ using the equations of motion and $W = Fs$.

Step-by-step Derivation:
1. Start with the equation of motion for uniform acceleration $a$, assuming the object starts from rest ($u=0$):
$$v^2 = u^2 + 2as \implies v^2 = 0 + 2as$$ $$\therefore as = \frac{1}{2}v^2$$
2. Apply Newton’s Second Law ($F = ma$):
$$a = \frac{F}{m}$$
3. Substitute the expression for $a$ into the motion equation ($as = \frac{1}{2}v^2$):
$$(\frac{F}{m})s = \frac{1}{2}v^2$$
4. Rearrange to find the work done $W = Fs$. The work done in accelerating the object from rest is equal to its final kinetic energy $E_k$:
$$Fs = E_k = \frac{1}{2}mv^2$$
Final Formula:
$$E_k = \frac{1}{2}mv^2$$

5.1 Power (P)

Definition of Power

Power (P) is defined as the rate at which work is done or the rate at which energy is transferred.

Key Formula:
$$P = \frac{W}{t}$$

Where:

• \(P\) is the power (measured in Watts, W).
• \(W\) is the work done or energy transferred (J).
• \(t\) is the time taken (s).

Units of Power

The SI unit for power is the Watt (W). One Watt is defined as one Joule of energy transferred per second.
$$1 \text{ W} = 1 \text{ J/s}$$

The Power-Force-Velocity Relationship: $P = Fv$

This is a highly useful formula, especially when dealing with vehicles or motors moving at constant speeds against resistive forces (like drag or friction).

Step-by-step Derivation (Required):
1. Start with the definition of power:
$$P = \frac{W}{t}$$
2. Substitute the formula for work done, $W = Fs$ (assuming force $F$ is constant and parallel to displacement $s$):
$$P = \frac{Fs}{t}$$
3. Recall that speed $v$ is displacement divided by time ($v = s/t$):
$$P = F (\frac{s}{t})$$
Final Formula:
$$P = Fv$$

Application Tip: If a car is moving at a constant velocity, the forward driving force provided by the engine must exactly match the total resistive force (air resistance + friction). If the car maintains constant velocity $v$, the power generated by the engine is simply $P = F_{\text{resistance}} \times v$.

5.1 Efficiency ($\eta$)

Defining and Calculating Efficiency

No machine or system is 100% perfect. Some energy is always "wasted," usually as heat or sound. Efficiency tells us how much of the energy input is turned into useful energy output.

Efficiency ($\mathbf{\eta}$) is the ratio of useful energy (or power) output to the total energy (or power) input.

Key Formulas:

Efficiency can be calculated using either energy or power:

1. Using Energy:
$$\text{Efficiency} = \frac{\text{Useful Energy Output}}{\text{Total Energy Input}} \times 100\%$$
2. Using Power (Work done per unit time):
$$\text{Efficiency} = \frac{\text{Useful Power Output}}{\text{Total Power Input}} \times 100\%$$

Example: A kettle draws 2000 J of electrical energy (Total Input) but only heats the water using 1800 J (Useful Output). The remaining 200 J is lost as heat to the air and sound.
$$\eta = \frac{1800 \text{ J}}{2000 \text{ J}} \times 100\% = 90\%$$

Key Takeaway: Efficiency is a unitless quantity (since the units cancel out) and is typically expressed as a percentage or a decimal value (always between 0 and 1).

Chapter Summary: Key Takeaways

Work, energy, and power are inextricably linked. Remember these core definitions and relationships:
* Work Done: $W = Fs$ (Force must be parallel to displacement).
* Energy Conservation: Energy is transferred, never lost (though some may be "wasted" into non-useful forms like heat).
* Kinetic Energy: $E_k = \frac{1}{2}mv^2$ (Energy of motion).
* Potential Energy: $\Delta E_p = mg\Delta h$ (Energy stored due to height in a uniform field).
* Power Definition: $P = W/t$ (Rate of doing work).
* Power-Velocity Link: $P = Fv$ (Essential for calculating forces required to maintain speed).