Hello Future Physicists! Welcome to Forces and Energy
Welcome to a really important chapter! We are moving from studying just forces (pushes and pulls) to understanding how those forces lead to energy transfer and movement. This chapter is the bridge that connects the mechanics you’ve already learned to the energy calculations we will use throughout the course.
Don't worry if the formulas look complicated at first! We will break down every concept step-by-step using simple language and relatable examples. Ready to dive into how things move and why? Let's go!
Quick Review: The Basics of Energy
Remember the Law of Conservation of Energy? It states that energy cannot be created or destroyed, only transferred from one store to another (or dissipated). This entire chapter is about calculating how much energy is being transferred or stored.
1. Work Done: The Physical Definition of Effort
What is Work Done?
In everyday life, 'work' means effort. But in Physics, Work Done has a very specific definition:
Work is done whenever a force causes an object to move in the direction of the force.
- If you push a wall as hard as you can, but it doesn't move, you did zero work (in physics terms!), even though you feel tired.
- If you lift a book, you are applying a force upwards and the book moves upwards. Work is done!
Calculating Work Done (\(W\))
The amount of work done depends on two things: the size of the force applied, and the distance the object moves.
Formula:
$$W = F \times d$$
Where:
- \(W\) is Work Done (or Energy Transferred), measured in Joules (J)
- \(F\) is the Force applied, measured in Newtons (N)
- \(d\) is the distance moved in the direction of the force, measured in metres (m)
Key Connection: Work Done is Energy Transferred
This is one of the most important takeaways!
Work Done = Energy Transferred (or used).
Because work is a measure of energy transfer, it is measured in the same unit: Joules (J).
Example Walkthrough
Imagine pushing a shopping cart with a constant force of 50 N over a distance of 10 m.
\(F = 50 \, \text{N}\)
\(d = 10 \, \text{m}\)
\(W = 50 \, \text{N} \times 10 \, \text{m} = 500 \, \text{J}\)
Key Takeaway: 500 J of energy was transferred to the shopping cart (mostly as kinetic energy, but some wasted as heat due to friction).
Work is the transfer of energy using a force over a distance. \(W = Fd\). Measured in Joules (J).
2. Gravitational Potential Energy (GPE)
Energy Stored by Height
Gravitational Potential Energy (GPE) is the energy stored in an object because of its position above the ground (its height) within a gravitational field.
Think of a brick resting high up on a shelf. It has GPE because if it falls, gravity will pull it down and convert that stored energy into kinetic energy (movement).
Calculating GPE
GPE depends on three things: the mass of the object, the gravitational field strength, and the height.
Prerequisite Check: Gravity on Earth
For calculations on Earth, the Gravitational Field Strength (\(g\)) is usually taken as \(9.8 \, \text{N/kg}\) (or sometimes rounded to \(10 \, \text{N/kg}\) depending on the exam paper instructions – always check the data sheet!).
Formula:
$$GPE = m \times g \times h$$
Where:
- \(GPE\) is Gravitational Potential Energy, measured in Joules (J)
- \(m\) is the mass of the object, measured in kilograms (kg)
- \(g\) is Gravitational Field Strength, measured in N/kg (or \(m/s^2\))
- \(h\) is the vertical height above the reference point, measured in metres (m)
Memory Aid for GPE
To remember the formula, think of "Mighty Good Height" (M G H)!
Common Mistake to Avoid!
Students sometimes use the object's weight instead of its mass. If they give you the weight in Newtons, you only need to multiply (Weight \(\times\) height). If they give you the mass in kilograms, you must include \(g\)!
A pumped hydro-electric power station uses GPE. It pumps water up into a high reservoir (storing huge amounts of GPE). When power is needed, the water is released, falling down and turning turbines, converting GPE into electrical energy!
3. Kinetic Energy (KE)
Energy of Movement
Kinetic Energy (KE) is the energy an object possesses because it is moving. Any object with mass and speed has kinetic energy.
When you throw a ball, the chemical energy in your muscles is converted into kinetic energy in the ball.
Calculating Kinetic Energy
Kinetic Energy depends on the object's mass and its speed (velocity).
Formula:
$$KE = \frac{1}{2} \times m \times v^2$$
Where:
- \(KE\) is Kinetic Energy, measured in Joules (J)
- \(m\) is the mass of the object, measured in kilograms (kg)
- \(v\) is the speed (velocity) of the object, measured in metres per second (m/s)
The Importance of Squaring the Velocity (\(v^2\))
Look closely at the formula: the velocity (\(v\)) is squared!
This means that speed has a much bigger effect on KE than mass does. If you double the mass, the KE doubles. But if you double the speed, the KE quadruples (\(2^2 = 4\)).
This is why driving faster is so dangerous: a small increase in speed results in a huge increase in the energy that must be lost (crashed) if the vehicle needs to stop suddenly.
Step-by-Step Calculation: KE
A 1000 kg car is traveling at 20 m/s. Calculate its KE.
1. Square the velocity: \(v^2 = 20 \times 20 = 400\)
2. Multiply mass and squared velocity: \(m \times v^2 = 1000 \times 400 = 400,000\)
3. Halve the result: \(KE = 0.5 \times 400,000 = 200,000 \, \text{J}\)
Energy Transfers in Action (KE and GPE)
On a roller coaster, energy constantly switches between KE and GPE:
- At the top of the highest hill: Maximum GPE, Minimum KE (\(v\) is slow).
- At the bottom of the steepest drop: Minimum GPE, Maximum KE (\(v\) is fast).
GPE: Stored due to height. \(GPE = mgh\).
KE: Stored due to speed. \(KE = \frac{1}{2}mv^2\).
4. Power: The Rate of Energy Transfer
Defining Power
If two machines do the exact same amount of work (transfer the same amount of energy), how do we compare them? We use Power.
Power is the rate at which work is done, or the rate at which energy is transferred.
A high-power device transfers a lot of energy very quickly. A low-power device might transfer the same total energy, but it takes much longer.
Unit of Power
Power is measured in Watts (W).
$$1 \, \text{Watt} = 1 \, \text{Joule of energy transferred per second} (1 \, \text{J/s})$$
Calculating Power (\(P\))
Since power is the rate of energy transfer, we divide the work done (or energy transferred) by the time taken.
Formula 1 (Using Work/Energy):
$$P = \frac{W}{t} \quad \text{or} \quad P = \frac{E}{t}$$
Where:
- \(P\) is Power, measured in Watts (W)
- \(W\) or \(E\) is Work Done or Energy Transferred, measured in Joules (J)
- \(t\) is the Time taken, measured in seconds (s)
Analogy: The Weightlifter
Two weightlifters lift a 100 kg weight 2 metres high.
*Work Done (W)*: They both do the same amount of work (\(W = Fd\)).
*Power (P)*: The weightlifter who lifts the weight in 1 second has more power than the one who takes 5 seconds, because they transferred the energy faster.
Alternative Power Calculation (Force and Speed)
Sometimes you don't know the time taken, but you know the force applied and the speed of the object. This is common for cars or motors traveling at a constant speed.
Formula 2 (Using Force and Velocity):
$$P = F \times v$$
Where:
- \(P\) is Power (W)
- \(F\) is the Force applied (N)
- \(v\) is the velocity or speed (m/s)
This formula is particularly useful when calculating the power needed to keep a car moving at a steady speed against friction and air resistance.
Power is how fast energy is used. \(P = E/t\). Use time in seconds!
5. Efficiency: How Well We Use Energy
Why Efficiency Matters
Due to the Law of Conservation of Energy, energy is never truly "lost." However, in any real-world process, some energy is always transferred to stores that are not useful (usually as heat, light, or sound). This is often called wasted energy or dissipated energy.
Efficiency measures how much of the total energy input is converted into useful energy output. We want high efficiency!
Calculating Efficiency
Efficiency is a ratio (a fraction) of useful energy output compared to total energy input. It can be expressed as a decimal (between 0 and 1) or as a percentage (between 0% and 100%).
Formula (as a decimal):
$$\text{Efficiency} = \frac{\text{Useful Output Energy (or Power)}}{\text{Total Input Energy (or Power)}}$$
Formula (as a percentage):
$$\text{Efficiency} (\%) = \frac{\text{Useful Output Energy}}{\text{Total Input Energy}} \times 100$$
Analogy: The Light Bulb
When you switch on a traditional (filament) light bulb:
- Total Input Energy: Electrical energy used (e.g., 100 J).
- Useful Output Energy: Light energy (e.g., 5 J).
- Wasted Output Energy: Heat energy (e.g., 95 J).
In this case, the efficiency is:
$$\text{Efficiency} = \frac{5 \, \text{J}}{100 \, \text{J}} \times 100 = 5\%$$
This is why modern LED bulbs (which produce much less heat) are much more efficient!
Understanding the Energy Flow
A helpful way to visualize energy flow and efficiency is using a Sankey Diagram (even if you don't have to draw them, understanding the principle is helpful):
Total Input Energy = Useful Output Energy + Wasted Energy
If you are asked to find the wasted energy, just subtract the useful output from the total input.
Important Rule about Efficiency
The efficiency calculated MUST always be less than or equal to 1 (or 100%). If your calculation gives you an efficiency of 150%, you know you have made a mistake! It is physically impossible to get more useful energy out than the total energy you put in.
Physics involves lots of steps, but remember to always check your units (kg, m, s) before starting any calculation. Good luck!