Welcome to the World of Work and Energy!
In this chapter, we are going to explore how objects move and what makes them do so. You might use the words "work" and "energy" every day, but in Oxford AQA Mathematics (9660), they have very specific meanings. Think of energy as "money" in a bank account and work as the "transaction" that moves that money around. By the end of these notes, you’ll be able to calculate exactly how much "effort" goes into moving objects and how they store that effort for later!
1. What is "Work Done"?
In maths, Work Done isn't about how hard you study! It happens only when a force moves an object over a distance. If you push as hard as you can against a brick wall and it doesn't move, you haven't done any "Work" in the mathematical sense, even if you are sweating!
The Basic Formula
Work is calculated using this simple equation:
\( W = Fs \)
Where:
• \( W \) is the Work Done (measured in Joules, J).
• \( F \) is the constant force applied (measured in Newtons, N).
• \( s \) is the distance moved (displacement) in the direction of the force (measured in metres, m).
Important: Direction Matters!
The force must be in the same direction as the movement. If you pull a suitcase at an angle, only the part of the force pulling it forward counts as work. Don't worry if this seems tricky; just remember that if the force and movement are perfectly aligned, you just multiply them!
Quick Review:
1 Joule is the work done when a force of 1 Newton moves an object 1 metre.
Key Takeaway: No movement = No work done!
2. Kinetic Energy (EK): The Energy of Motion
Anything that moves has Kinetic Energy. The faster it moves, or the heavier it is, the more kinetic energy it has. Think of a ping-pong ball versus a bowling ball moving at the same speed—the bowling ball has much more "energy" because it has more mass.
The Formula
\( E_k = \frac{1}{2}mv^2 \)
Where:
• \( m \) is the mass of the object in kg.
• \( v \) is the speed of the object in \( ms^{-1} \).
Did you know?
Because the speed \( v \) is squared, doubling your speed actually quadruples your kinetic energy! This is why car crashes at high speeds are so much more dangerous than at low speeds.
Key Takeaway: Kinetic energy depends on how heavy an object is and how fast it’s going. If an object is at rest (\( v = 0 \)), its kinetic energy is zero.
3. Gravitational Potential Energy (EP)
Potential Energy is "stored" energy. When you lift an object up against gravity, you are doing work on it, and that work is stored as Gravitational Potential Energy. If you let go, that stored energy turns into movement!
The Formula
\( E_p = mgh \)
Where:
• \( m \) is the mass (kg).
• \( g \) is the acceleration due to gravity (on Earth, we use \( 9.8 \, ms^{-2} \)).
• \( h \) is the vertical height it has been raised (m).
Analogy: Imagine a roller coaster at the very top of the first hill. It isn't moving fast yet, but it has a huge amount of \( E_p \) ready to be released as it zooms down!
Key Takeaway: Height is the hero here. The higher you go, the more potential energy you store.
4. The Conservation of Energy
This is one of the most important rules in all of science: Energy cannot be created or destroyed, only changed from one form to another.
In a perfect world (where there is no friction or air resistance), the total energy stays the same:
\( Initial \, (E_k + E_p) = Final \, (E_k + E_p) \)
Step-by-Step Example:
If you drop a ball from a height:
1. At the top, it has maximum \( E_p \) and zero \( E_k \).
2. As it falls, it loses height (loses \( E_p \)) but gains speed (gains \( E_k \)).
3. Just before it hits the ground, all that \( E_p \) has turned into \( E_k \)!
Common Mistake to Avoid:
Always check if there is friction. If a surface is "rough," some energy will be "lost" as heat. In this case:
Initial Energy = Final Energy + Work Done against Friction.
Key Takeaway: Total energy at the start must equal total energy at the end, as long as you account for heat and friction.
5. Power: How Fast Can You Work?
Power is simply the rate at which work is done. If two people climb the same stairs, they do the same amount of work, but the person who runs up has more Power because they did the work faster.
The Formulas
1. \( P = \frac{W}{t} \) (Work divided by time)
2. \( P = Fv \) (Force multiplied by velocity)
The unit for Power is the Watt (W). 1 Watt = 1 Joule per second.
Memory Aid:
Think of a lightbulb. A 100W bulb uses energy faster than a 40W bulb. In mechanics, a high-power engine can provide a large force even when the car is moving quickly (\( P = Fv \)).
Key Takeaway: Power is all about time. Doing the same job in half the time requires double the power!
Final Encouragement
Mechanics can feel like a lot of formulas at first, but remember that they all connect! Work changes Energy, and Power measures how fast that change happens. Practice identifying what information the question gives you (Mass? Height? Speed?), and you’ll find the right formula in no time. You've got this!