Newton’s Laws of Motion

Introduction

This section explores Newton's three laws of motion, fundamental principles in classical mechanics. We will examine the relationship between force, mass, and acceleration, and how these concepts apply to connected particles.

Newton's First Law: Inertia

Statement: An object at rest stays at rest and an object in motion stays in motion with the same speed and in the same direction unless acted upon by an external force.

Concept: This law introduces the concept of inertia, the tendency of an object to resist changes in its state of motion. Inertia is directly proportional to the mass of the object.

Mathematical Representation: This law is the foundation for understanding free-body diagrams and analyzing motion in the absence of net forces.

Example: A hockey puck sliding across the ice will continue to slide until friction (an external force) stops it.

Newton's Second Law: Force, Mass, and Acceleration

Statement: The net force acting on an object is equal to the mass of the object multiplied by its acceleration.

Mathematical Representation: $$F_{net} = ma$$

Variables:

• F_net: Net force (measured in Newtons, N)
• m: Mass (measured in kilograms, kg)
• a: Acceleration (measured in meters per second squared, m/s²)

Relationship: This law quantifies the relationship between force, mass, and acceleration. The larger the mass, the greater the force required to produce the same acceleration. The larger the force, the greater the acceleration.

Free-Body Diagrams: A crucial application of Newton's Second Law is in drawing free-body diagrams. These diagrams represent all the forces acting on an object, allowing us to determine the net force.

Example: A 10 kg box is pushed with a force of 50 N. The acceleration of the box is: $$a = \frac{F}{m} = \frac{50}{10} = 5 \, m/s^2$$

Newton's Third Law: Action and Reaction

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

Concept: Forces always occur in pairs. The action force and the reaction force act on different objects.

Mathematical Representation: $$F_{12} = -F_{21}$$

Example: When you push against a wall, the wall pushes back on you with an equal and opposite force. This is why you feel the wall.

Important Note: The action and reaction forces are not equal and opposite in terms of their effect on the same object. They act on different objects.

Connected Particles

Concept: When particles are connected by a string or rod, the forces acting on them must be considered as a system. We can use Newton's Second Law to analyze the motion of the connected particles.

Method:

1. Draw a free-body diagram for each particle.
2. Identify all the forces acting on each particle (e.g., tension, gravity).
3. Apply Newton's Second Law to each particle, writing equations for the translational and rotational motion (if applicable).
4. Solve the system of equations to find the unknown forces or accelerations.

Example: Two masses are connected by a string over a frictionless pulley. If one mass is heavier than the other, it will accelerate downwards, and the lighter mass will accelerate upwards. The tension in the string will be the same throughout the string.

Law	Statement	Mathematical Representation	Key Concepts
First Law (Inertia)	An object at rest stays at rest...	$$F_{net} = ma$$	Inertia, Free-Body Diagrams
Second Law	The net force acting on an object...	$$F_{net} = ma$$	Force, Mass, Acceleration, Free-Body Diagrams
Third Law (Action-Reaction)	For every action...	$$F_{12} = -F_{21}$$	Action-Reaction Pairs
Connected Particles	Forces on connected particles must be analyzed as a system.	Newton's Second Law applied to each particle.	Free-Body Diagrams, System Analysis