Physics (9630) | Newton’s laws of motion

Newton’s Laws of Motion: The Foundation of Mechanics (9630: Section 3.2.5)

Hello future physicists! Welcome to one of the most fundamental and crucial topics in all of science: Newton's Laws of Motion.
These three laws, developed by Sir Isaac Newton over 300 years ago, explain how and why objects move (or don't move!) in the world around us—from kicking a football to launching a satellite.
Mastering this chapter will unlock your understanding of forces, acceleration, and the entire section of Mechanics. Don't worry if the concepts seem simple yet profound; we'll break them down step-by-step!

Quick Review: Forces, Vectors, and Equilibrium

Before diving into the laws, remember that force is a vector quantity. This means it has both magnitude (size) and direction.

• Force (F): Measured in Newtons (N). A push or a pull.
• Mass (m): A measure of an object's resistance to acceleration (inertia). Measured in kilograms (kg). (This is a scalar quantity).
• Weight (W): The gravitational force acting on an object. \(W = mg\).

Equilibrium and Net Force

In physics, the key concept for applying Newton's laws is the Net Force (\(F_{net}\) or \(\Sigma F\)). This is the overall, resultant force acting on an object.
An object is in equilibrium when the net force acting on it is zero (\(\Sigma F = 0\)). This occurs under two specific conditions (Syllabus 3.2.1):

1. The object is at rest (stationary).
2. The object is moving with a constant velocity (constant speed in a straight line).

Key Takeaway: If all forces cancel out (zero net force), the object is in equilibrium, meaning its motion won't change.

1. Newton's First Law: The Law of Inertia (The 'Do Nothing' Law)

Newton’s First Law describes what happens when there is no net force acting on an object.

An object remains at rest, or continues to move at a constant velocity, unless acted upon by a resultant external force.

This law introduces the concept of Inertia.

What is Inertia?

Inertia is the tendency of an object to resist changes in its state of motion. Essentially, massive objects are 'lazy' and harder to start moving, stop moving, or change direction.
The amount of inertia an object has is directly related to its mass. A truck has much more inertia than a bicycle.

Real-World Example & Analogy

Imagine you are standing on a bus.

• When the bus starts suddenly: Your feet are dragged forward by the friction from the bus floor, but your upper body wants to maintain its original state of rest (inertia), so you lurch backward.
• When the bus brakes suddenly: Your feet stop, but your body wants to maintain its forward velocity (inertia), so you lurch forward.

Common Mistake to Avoid: Many students mistakenly think that constant velocity means no forces are acting. This is wrong! It means the net force is zero, as the forces balance out (i.e., it is in equilibrium).

Key Takeaway: The First Law defines equilibrium: \(\Sigma F = 0\) means acceleration \(a = 0\).

2. Newton's Second Law: Force, Mass, and Acceleration (The 'Action' Law)

The Second Law is the most crucial for calculations and describes what happens when a net force does act on an object. It connects force, mass, and motion change.

The Fundamental Definition (Syllabus 3.2.6)

The most fundamental way to state Newton's Second Law involves momentum (\(p\)):

The net force (\(F\)) acting on an object is directly proportional to the rate of change of momentum (\(\frac{\Delta p}{\Delta t}\)) of that object, and occurs in the direction of the net force.

Mathematically, this is expressed as: $$F = \frac{\Delta (mv)}{\Delta t}$$

The Simplified Equation (\(F = ma\)) (Syllabus 3.2.5)

In the situations you will usually encounter (where the mass (m) is constant), we can simplify this.
Since mass is constant, we can move it outside the change (\(\Delta\)): $$F = m \frac{\Delta v}{\Delta t}$$ Since \(\frac{\Delta v}{\Delta t}\) is the definition of acceleration (\(a\)), we arrive at the famous equation:

$$F = ma$$

Where:

• \(F\) is the Net Force (N)
• \(m\) is the Mass (kg)
• \(a\) is the Acceleration (\(\text{m s}^{-2}\))

This equation tells us two things:

1. For a constant mass, a larger net force produces a larger acceleration (\(F \propto a\)).
2. For a constant force, a larger mass produces a smaller acceleration (\(m \propto 1/a\)).

Step-by-Step Problem Solving using \(F=ma\)

1. Draw a Free-Body Diagram: Sketch the object and draw all forces acting on it (Weight, Tension, Friction, Thrust, Normal Contact Force, etc.).
2. Choose a Direction: Decide which direction is positive (usually the direction of acceleration or motion).
3. Calculate Net Force: Sum all forces in that chosen direction. Remember forces acting opposite to the positive direction are negative.
4. Apply the Law: Set the Net Force equal to \(ma\).

Example: A car with a driving force \(F_D\) of 2000 N experiences air resistance \(F_R\) of 500 N. If the car mass is 1000 kg, find acceleration \(a\).

$$F_{net} = F_D - F_R$$ $$F_{net} = 2000 \text{ N} - 500 \text{ N} = 1500 \text{ N}$$ $$F_{net} = ma \implies 1500 = (1000) a$$ $$a = 1.5 \text{ m s}^{-2}$$

Did you know? A force of 1 Newton is defined as the force required to accelerate a 1 kg mass at 1 \(\text{m s}^{-2}\).

Impulse and Contact Time (Syllabus 3.2.6)

Rearranging the momentum definition gives us the concept of Impulse: $$F \Delta t = \Delta (mv)$$

The quantity \(F\Delta t\) (Force multiplied by the time over which it acts) is called the Impulse. Impulse is equal to the change in momentum (\(\Delta p\)).

For a constant force \(F\), this relationship shows the link between impact forces and contact times (e.g., kicking a football, crumple zones):

If the change in momentum (\(\Delta p\)) required to stop an object is fixed, then to reduce the force (F), you must increase the time of contact (\(\Delta t\)): $$F = \frac{\Delta p}{\Delta t}$$

Analogy: Car Safety. Crumple zones in cars increase the contact time (\(\Delta t\)) during a collision, dramatically reducing the huge force (F) exerted on the occupants, thus minimizing injury.

Key Takeaway: The Second Law dictates motion change: \(F_{net} = ma\). Impulse (\(F\Delta t\)) equals the momentum change.

3. Newton's Third Law: Action and Reaction (The 'Pair' Law)

The Third Law describes the nature of forces themselves—they always come in pairs.

Whenever object A exerts a force on object B, object B simultaneously exerts an equal and opposite force on object A.

Mathematically: $$F_{A \text{ on } B} = -F_{B \text{ on } A}$$

Key Characteristics of Third Law Pairs

• The forces are equal in magnitude.
• The forces are opposite in direction.
• The forces act on different bodies. (This is the most important distinction!)
• The forces are of the same type (e.g., gravitational, electrical, normal).

Analogy: Walking

When you walk, you push the Earth backward with your foot (Action). The Earth, in response, pushes you forward with an equal and opposite force (Reaction). This reaction force is what accelerates you forward.

Common Mistake: Confusing 3rd Law Pairs with Equilibrium

The forces in a Third Law pair never cancel out because they act on different objects.

Example: A book resting on a table.

1. Force Pair 1 (Gravitational):
   • Action: Earth pulls the book down (Weight, \(W\)).
   • Reaction: Book pulls the Earth up (Equal gravitational force).
2. Force Pair 2 (Contact):
   • Action: Book pushes down on the table (Force on Table, \(F_B\)).
   • Reaction: Table pushes up on the book (Normal Contact Force, \(R\)).

The forces \(W\) and \(R\) act on the same object (the book). If \(W=R\), the book is in equilibrium (First Law), but they are not a Third Law pair.

Key Takeaway: Forces come in equal and opposite pairs acting on different objects.

Applications of Newton's Laws (Syllabus 3.2.4)

Newton's laws are used to analyze all types of motion, particularly when resistive forces like friction and air resistance are involved.

A. Friction and Drag (Qualitative Treatment)

Friction (when sliding along a surface) and Drag/Air Resistance (when moving through a fluid like air or water) are forces that oppose motion.

• Qualitative Treatment: You need to understand that these forces exist and oppose the applied force, reducing the net force and therefore the acceleration.
• Key Rule: Air resistance increases as the speed of the object increases.

B. Terminal Speed (Velocity)

Terminal speed is the highest speed an object can reach when falling through a fluid (like air). It is a perfect example of applying both the First and Second Laws.

Step-by-step process for reaching terminal speed:

1. Start (Time \(t=0\)): The object starts falling. Speed is zero, so drag is zero. The only downward force is Weight (\(W\)). $$F_{net} = W$$ Acceleration \(a = W/m\) (this is maximum acceleration, \(g\)).
2. Intermediate Stage: Speed increases, so drag (\(D\)) increases. $$F_{net} = W - D$$ Since \(D\) is growing, \(F_{net}\) decreases, and therefore acceleration (\(a\)) decreases.
3. Terminal Speed: The object continues to accelerate until the drag force (\(D\)) becomes equal to the weight (\(W\)). $$F_{net} = W - D = 0$$ Since \(F_{net} = 0\), the object is in equilibrium (First Law) and moves at a constant, maximum speed known as Terminal Speed (\(v_T\)). Acceleration is zero.

Analogy: A skydiver accelerates rapidly when they jump out of a plane, but eventually, the air resistance pushing up equals their weight pulling down. They then fall at a constant terminal velocity until they open the parachute (which massively increases the drag force).

Quick Review Box: Newton’s Laws

• First Law (Inertia): If \(F_{net}=0\), then \(a=0\) (Equilibrium).
• Second Law (\(F=ma\)): If \(F_{net} \neq 0\), then \(F_{net} = ma\). This force causes acceleration in the direction of \(F_{net}\).
• Third Law (Action/Reaction): Forces always occur in equal and opposite pairs acting on different objects.