Newton’s laws for a particle

Introduction to Newton’s Laws

Welcome to one of the most exciting parts of Mechanics! Have you ever wondered why you lurch forward when a bus suddenly brakes, or why it’s harder to push a full shopping trolley than an empty one? These everyday moments are all governed by three simple rules discovered by Sir Isaac Newton. In this chapter, we will learn how to turn these observations into mathematical equations to predict how objects move.

Newton’s First Law: The Law of Inertia

Newton’s First Law states: An object continues in a state of rest or uniform motion in a straight line unless it is acted on by a resultant force.

In simple terms, objects are "lazy." If they are sitting still, they want to stay still. If they are moving at a steady speed in a straight line, they want to keep doing exactly that forever.

What is a Resultant Force?

A resultant force (often called the net force) is the single force you get when you combine all the different forces acting on an object.
- If the forces are balanced, the resultant force is zero. The object’s velocity won't change.
- If the forces are unbalanced, there is a resultant force, and the object will speed up, slow down, or change direction.

Quick Review:
- Zero Resultant Force = Stationary OR Constant Velocity.
- Non-zero Resultant Force = Acceleration.

Newton’s Second Law: The "Workhorse" Equation

This is the law you will use most often in your exams. It links force, mass, and acceleration.

Newton’s Second Law states: A resultant force \(F\) acting on an object of fixed mass \(m\) gives the object an acceleration \(a\) given by \(F = ma\).

Breaking Down the Formula

\(F = ma\)

- \(F\) is the resultant force measured in Newtons (N).
- \(m\) is the mass of the particle in kilograms (kg).
- \(a\) is the acceleration in \(ms^{-2}\).

Did you know? In vector form, this is written as \(\mathbf{F} = m\mathbf{a}\). This means the acceleration always happens in the same direction as the resultant force.

Weight vs. Mass

Don't let these two trip you up!
- Mass is how much "stuff" is in an object (measured in kg). It stays the same everywhere.
- Weight is a force caused by gravity. On Earth, we calculate weight using \(W = mg\), where \(g\) is the acceleration due to gravity (usually \(9.8 \ ms^{-2}\) in MEI Mathematics B).

Key Takeaway: Always use the resultant force in \(F = ma\). If a car has a driving force of \(1000 \ N\) and air resistance of \(200 \ N\), the \(F\) you use is \(1000 - 200 = 800 \ N\).

Newton’s Third Law: Pairs of Forces

Newton’s Third Law states: When one object exerts a force on another, there is always a reaction which is equal in magnitude and opposite in direction to the acting force.

Think of this as the "Mirror Law." If you push a wall with \(50 \ N\) of force, the wall pushes back on you with exactly \(50 \ N\).

Important Tip for N3L

These forces always act on different objects. If you are sitting on a chair:
1. You push down on the chair.
2. The chair pushes up on you (this is called the Normal Reaction Force).

Formulating the Equation of Motion

An equation of motion is just the \(F = ma\) formula filled in with the specific forces acting on your particle.

Step-by-Step: How to Solve Problems

1. Draw a Diagram: Use a simple dot or box to represent the particle.
2. Label all Forces: Include weight (\(mg\)), tension (\(T\)), friction (\(f\)), and normal reaction (\(R\)).
3. Choose a Direction: Decide which way is positive (usually the direction of acceleration).
4. Find the Resultant Force (\(F\)): Subtract the forces going the "wrong" way from the forces going the "right" way.
5. Apply \(F = ma\).

Example: A box of mass \(5 \ kg\) is pulled along a smooth floor by a force of \(20 \ N\).
Equation: \(20 = 5a\)
Result: \(a = 4 \ ms^{-2}\)

Connected Particles

Sometimes particles are joined together, like a car towing a caravan or two weights over a pulley.

The System Approach

If two particles are moving together in a straight line (no relative motion), you can treat them as a single particle with a mass equal to the total mass of the system.
Example: A locomotive (\(1000 \ kg\)) pulling a carriage (\(500 \ kg\)) can be treated as one \(1500 \ kg\) object to find the overall acceleration.

Internal vs. External Forces

- External Forces: Forces from outside the system (like the engine’s driving force or friction).
- Internal Forces: Forces acting between the parts (like the Tension in the rope connecting them).
Memory Trick: When you look at the whole system together, internal forces (Tension) "cancel out" because they pull equally in opposite directions!

Pulleys

In the MEI syllabus, we usually assume pulleys are smooth and strings are light and inextensible.
- Light: We ignore the mass of the string.
- Inextensible: The string doesn't stretch, so both particles must have the same acceleration.

Common Mistake: Forgetting that tension (\(T\)) pulls away from the particle at both ends of the string. When setting up equations for connected particles, always write a separate \(F = ma\) equation for each particle first.

Quick Summary Checklist

- Have you identified the resultant force?
- Is the mass in kg?
- Did you include gravity (\(g = 9.8\)) if the motion is vertical?
- If there's no acceleration, did you set \(F = 0\) (Equilibrium)?
- Don't worry if it seems tricky at first—mechanics is all about practice and drawing clear diagrams!