Welcome to Mechanics: Understanding How Things Move!
Hello future Physicists! Mechanics might sound like a massive, tricky topic, but it’s actually the study of how things move, why they move, and what happens when they collide—which means it’s all around us! From driving a car to launching a rocket, mechanics explains it all.
Don’t worry if some concepts seem challenging at first. We will break them down step-by-step, using simple language and real-world examples. By the end of these notes, you will have the foundation needed to conquer your exams!
Section 1: The Language of Motion – Scalars and Vectors
Before we calculate anything, we need to know the difference between two fundamental types of quantities used in Physics.
1.1 Scalars (The Simple Numbers)
A Scalar Quantity is defined completely by its magnitude (size) only. It does not have a direction.
Examples: Distance, Speed, Mass, Time, Energy, Temperature.
Quick Tip: If you answer a question about a scalar, just a number and a unit is enough!
1.2 Vectors (Magnitude AND Direction)
A Vector Quantity requires both magnitude and a specific direction to be fully described.
Examples: Displacement, Velocity, Acceleration, Force, Momentum.
Analogy Alert!
Imagine telling someone how long you ran:
"I ran 5 kilometres." (This is distance - a scalar.)
"I ran 5 kilometres east." (This is displacement - a vector.)
1.3 Adding Vectors
Because vectors have direction, we can't just add them up like regular numbers unless they point along the same line.
For two vectors acting at right angles (90°), we use Pythagoras’ theorem and trigonometry to find the Resultant Vector (the single vector that represents the combined effect).
- Magnitude: Use Pythagoras: \(R^2 = A^2 + B^2\)
- Direction (\(\theta\)): Use trigonometry: \(\tan \theta = \frac{\text{Opposite}}{\text{Adjacent}}\)
Key Takeaway: Always check if the quantity is a vector or a scalar. If it's a vector, you must state the direction!
Section 2: Kinematics – Describing Motion
Kinematics is the branch of mechanics that describes motion without reference to the forces causing it. We use precise terms to replace everyday words.
2.1 Key Kinematic Definitions
- Displacement (\(s\)): Change in position of an object (Vector). Unit: metres (m).
- Velocity (\(v\)): Rate of change of displacement (Vector). Unit: \(\text{m s}^{-1}\).
- Acceleration (\(a\)): Rate of change of velocity (Vector). Unit: \(\text{m s}^{-2}\).
Did you know? If an object is moving in a circle at a constant speed, its velocity is constantly changing because its direction is changing. Therefore, it is accelerating!
2.2 Constant Acceleration (SUVAT Equations)
The SUVAT equations are your best friends when dealing with motion under constant acceleration (like a car braking, or something falling). Remember, these equations only work if acceleration is constant!
Let's define the variables:
- S = Displacement (m)
- U = Initial Velocity (\(\text{m s}^{-1}\))
- V = Final Velocity (\(\text{m s}^{-1}\))
- A = Constant Acceleration (\(\text{m s}^{-2}\))
- T = Time taken (s)
The Four SUVAT Equations:
- \(v = u + at\) (Missing S)
- \(s = ut + \frac{1}{2}at^2\) (Missing V)
- \(v^2 = u^2 + 2as\) (Missing T)
- \(s = \frac{(u+v)}{2}t\) (Missing A)
How to Use SUVAT:
- List the variables (S, U, V, A, T) and fill in the known values.
- Note which variable is unknown and which variable is missing from your list.
- Choose the equation that uses the variables you have and need.
- Crucial Rule: Always define a positive direction (e.g., up is positive). Any vector (displacement, velocity, acceleration) pointing the opposite way must be given a negative sign!
2.3 Free Fall (Acceleration due to Gravity)
When an object is thrown or dropped, the only force (ignoring air resistance) acting on it is gravity. The acceleration caused by gravity is denoted by \(g\).
Key Value: \(g \approx 9.81 \, \text{m s}^{-2}\) (in calculations, use the value specified by your examination board, often 9.8 or 9.81).
If you throw a ball straight up:
The ball slows down as it rises (negative acceleration: \(a = -g\)).
At the highest point, the instantaneous velocity (\(v\)) is zero.
It speeds up as it falls (positive acceleration: \(a = +g\), if 'down' is your positive direction).
Section 3: Forces and Dynamics (Newton's Laws)
Dynamics explains why objects move. The answer is always: Force. A force is a push or a pull, measured in Newtons (N).
3.1 Newton’s First Law: The Law of Inertia
An object will remain at rest or continue to move at a constant velocity (zero acceleration) unless acted upon by a Net (or Resultant) Force.
In simpler terms: If all the forces acting on an object balance out (resultant force is zero), the object won't accelerate. It will either be motionless or moving steadily. This state is called Equilibrium.
3.2 Newton’s Second Law: F = ma
The acceleration (\(a\)) of an object is directly proportional to the net force (\(F\)) acting on it and inversely proportional to its mass (\(m\)).
The Most Important Equation in Dynamics: $$F = ma$$ where \(F\) is the resultant force acting on the object (measured in Newtons).
Common Mistake to Avoid: When using \(F=ma\), the \(F\) must be the net force. If a box is being pushed with 10 N and friction opposes with 2 N, the net force \(F\) is \(10 - 2 = 8 \, \text{N}\).
3.3 Newton’s Third Law: Action and Reaction
When two objects interact, they exert equal and opposite forces on each other.
Key Points of the Third Law Pair:
- They are always equal in magnitude and opposite in direction.
- They must act on different objects.
- They must be of the same type (e.g., both gravity, or both contact forces).
Example: If you push a wall with 50 N (Action), the wall pushes back on you with 50 N (Reaction). You push the wall; the wall pushes you. They are acting on different bodies.
3.4 Weight and Mass
Mass (\(m\)): A measure of the amount of matter in an object. It is a scalar and remains constant everywhere. Unit: kg.
Weight (\(W\)): The force of gravity acting on a mass. It is a vector and changes depending on the gravitational field strength (\(g\)). Unit: Newtons (N).
The Weight Equation: $$W = mg$$ where \(g\) is the gravitational field strength (\(\text{N kg}^{-1}\) or \(\text{m s}^{-2}\)).
3.5 Drag and Terminal Velocity
Drag (or air resistance) is a force that opposes motion through a fluid (liquid or gas). It increases as the object's speed increases.
Terminal Velocity: When an object falls, gravity pulls it down, and drag pulls it up.
- Starts falling: Weight > Drag, so the object accelerates.
- Speed increases: Drag increases significantly.
- Forces balance: Eventually, Drag force equals Weight force. The Net Force is Zero.
- Constant speed: The acceleration stops, and the object continues falling at its maximum, constant speed, called Terminal Velocity (obeying Newton's First Law).
Key Takeaway: Newton's second law is the backbone of dynamics. If acceleration is non-zero, there must be a resultant force causing it.
Section 4: Energy, Work, and Power
These concepts are crucial because they allow us to analyze motion without focusing directly on time or acceleration.
4.1 Work Done
Work Done (\(W\)) is done when a force causes an object to move in the direction of the force. It transfers energy.
$$W = F d \cos \theta$$
where \(F\) is the force, \(d\) is the distance moved, and \(\theta\) is the angle between the force and the displacement.
If the force is applied parallel to the displacement, \(\cos \theta = 1\), and:
$$W = Fd$$
Unit: Joule (J). (1 J = 1 N m).
Example: Lifting a box vertically does work. Carrying it horizontally across a room at constant velocity does zero work against gravity, because the gravitational force is perpendicular to the displacement.
4.2 The Principle of Conservation of Energy
Energy cannot be created or destroyed; it can only be transferred from one form to another.
4.3 Key Forms of Mechanical Energy
1. Gravitational Potential Energy (GPE), \(E_p\): Energy stored in an object due to its height (position in a gravitational field). $$E_p = mgh$$
2. Kinetic Energy (KE), \(E_k\): Energy an object possesses due to its motion. $$E_k = \frac{1}{2}mv^2$$
4.4 Power and Efficiency
Power (\(P\)) is the rate at which work is done or the rate at which energy is transferred.
$$P = \frac{W}{t} = \frac{E}{t}$$ Unit: Watt (W). (1 W = 1 J/s).
Power related to Force and Velocity: If a constant force \(F\) moves an object at a constant velocity \(v\): $$P = Fv$$ (This is useful for vehicles moving at steady speeds against constant drag forces).
Efficiency: A measure of how well energy input is converted into useful output. $$\text{Efficiency} = \frac{\text{Useful Output Energy (or Power)}}{\text{Total Input Energy (or Power)}} \times 100\%$$
Because some energy is always lost (usually as heat or sound), efficiency is always less than 100%.
Section 5: Momentum and Collisions
Momentum helps us analyse complex interactions like collisions or explosions.
5.1 Defining Momentum
Momentum (\(p\)) is the mass of an object multiplied by its velocity. It is a vector quantity.
$$p = mv$$ Unit: \(\text{kg m s}^{-1}\).
5.2 The Principle of Conservation of Momentum
In a closed system (where no external forces act), the total momentum before a collision or explosion is equal to the total momentum after the event.
$$m_1 u_1 + m_2 u_2 = m_1 v_1 + m_2 v_2$$
Crucial Reminder: Since momentum is a vector, you must be extremely careful with signs. Define one direction as positive (e.g., right = positive) and the opposite direction as negative.
5.3 Types of Collisions
In all collisions, momentum is conserved. However, kinetic energy may or may not be conserved:
- Elastic Collision: Both momentum and kinetic energy are conserved. (Rare in the real world, often used as an ideal model).
- Inelastic Collision: Momentum is conserved, but kinetic energy is not conserved (some is converted into heat, sound, or permanent deformation).
- Perfectly Inelastic: The objects stick together after the collision (e.g., a bullet embedding itself in wood). This is the case where the maximum possible KE is lost.
5.4 Impulse (Force and Time)
From Newton's Second Law, \(F = ma = m \frac{(v-u)}{t}\). Rearranging gives us:
$$Ft = m(v-u)$$
The term \(m(v-u)\) is the change in momentum (\(\Delta p\)).
The term \(Ft\) is the Impulse, which is defined as the product of the force and the time interval over which it acts.
$$\text{Impulse} = \text{Change in Momentum}$$
Real-World Connection: Airbags and crumple zones in cars are designed to increase the time (\(t\)) taken for a collision. If the change in momentum (\(\Delta p\)) is fixed, increasing \(t\) must decrease the force (\(F\)), reducing injury.
Chapter Summary: Mechanics Core Concepts
You have now mastered the fundamental tools of motion, forces, and energy!
Final Check List:
- Can you distinguish between scalar and vector quantities?
- Can you select and apply the correct SUVAT equation?
- Do you know that \(F=ma\) uses the net force?
- Do you understand that the 3rd Law pairs act on different bodies?
- Can you use the conservation of energy (\(E_k\) and \(E_p\))?
- Are you prepared to handle direction (signs) when calculating momentum?
Keep practicing those problem-solving techniques! You've got this!
-- End of Mechanics Study Notes --