3.10 Newton's Laws of Motion

First formulated by Sir Isaac Newton in his 1687 masterpiece Principia Mathematica, Newton's three laws of motion are the foundational principles of classical mechanics. These laws describe the relationship between the motion of an object and the forces acting upon it.

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1. Newton's First Law of Motion (The Law of Inertia)

An object will remain at rest, or continue to move with a constant velocity, unless acted upon by a net external force.

This law essentially states that objects have a natural tendency to resist changes in their state of motion.

Inertia

Inertia is the inherent property of an object that makes it resist any change in its state of rest or uniform motion.

The mass of an object is a quantitative measure of its inertia. The more massive an object is, the greater its inertia, and the harder it is to change its state of motion. For example, it is much easier to push a bicycle than a car because the car has more mass and therefore more inertia.

Frame of Reference

Newton's laws are only valid in a specific type of coordinate system called a frame of reference.

Inertial Frame of Reference: A frame of reference that is not accelerating (at rest or moving with constant velocity). In an inertial frame, Newton's First Law holds true — an object with no net force will not accelerate.

Non-Inertial Frame of Reference: A frame of reference that is accelerating (e.g., an accelerating car, a spinning merry-go-round, a braking train). In a non-inertial frame, Newton's laws do not appear to hold without introducing fictitious (pseudo) forces.

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2. Newton's Second Law of Motion

The acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. The direction of the acceleration is in the direction of the net force.

Mathematical Formulation

$\vec{a}\propto {\vec{F}}_{net}\text{and}\vec{a}\propto \frac{1}{m}$

Combining these proportionalities and choosing the Newton as the unit of force:

$\boxed{{\vec{F}}_{net}=m\vec{a}}$

This is a vector equation — the net force and acceleration are always in the same direction.

SI Unit of Force

The SI unit of force is the Newton (N):

$1\text{ N}=1\text{ kg}\cdot \text{m}\cdot {\text{s}}^{-2}$

Newton's Second Law in Terms of Momentum

Newton originally stated his second law in terms of the rate of change of momentum:

$\vec{F}=\frac{\Delta \vec{p}}{\Delta t}$

For constant mass, $\Delta \vec{p}=m\Delta \vec{v}=m\vec{a}\Delta t$, which reduces to $\vec{F}=m\vec{a}$.

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3. Newton's Third Law of Motion

For every action, there is an equal and opposite reaction.

Key Characteristics

• Forces occur in pairs: If object A exerts a force on object B (action), then object B simultaneously exerts a force on object A (reaction).
• Equal in magnitude: $|{\vec{F}}_{AB}|=|{\vec{F}}_{BA}|$
• Opposite in direction: ${\vec{F}}_{AB}=-{\vec{F}}_{BA}$
• Act on different objects: The action and reaction forces act on different objects — they do not cancel each other out.

Example: When you push against a wall (action), the wall pushes back on you with an equal and opposite force (reaction). Your push acts on the wall; the wall's push acts on you.

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Mass vs. Weight

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Summary Table