LA_296_Module 3_Forces, Mass, and Newton's Laws: Foundations of Classical Mechanics

Module Learning Objectives

• Understand Newton's Laws of Motion: Demonstrate understanding of Newton's laws and the concepts of force and mass.

• Gravitational Force: Specify the behavior of the force of gravity on Earth according to Newton's laws.

• Historical Perspectives: Compare the principles of mechanics as held by Aristotle, Galileo, and Newton.

• Common Forces: Demonstrate understanding of common forces, including universal gravity, elastic forces, normal forces, and friction forces.

• Real-World Applications: Explain how these forces commonly occur in the real world.

Readings

• Douglas Giancoli, Physics: Principles with Applications Volume 1:

  • Chapter 4, Sections 4.1–4.6

  • Chapter 5, Sections 5.4–5.7

  • Chapter 3, Sections 3.5–3.8

Foundations of Force and Motion

Historical Development of Mechanics

Classical mechanics, the study of motion and forces, has evolved through the contributions of Aristotle, Galileo, and Newton. Each provided key insights into how and why objects move.

• Aristotle: Believed a force was required to keep an object moving. He thought the natural state of objects was to be at rest.

• Galileo: Demonstrated that objects in motion remain in motion unless acted upon by an external force (inertia). He also showed that all objects fall at the same rate in the absence of air resistance.

• Newton: Formulated the three laws of motion, providing a mathematical framework for understanding forces and motion.

Example: A ball rolling on a smooth surface will keep moving at constant speed unless friction or another force slows it down.

Types of Forces

Forces are pushes or pulls that can cause changes in an object's motion. They are vector quantities, meaning they have both magnitude and direction.

• Contact Forces: Require physical contact (e.g., friction, tension, normal force).

• Non-contact Forces: Act at a distance (e.g., gravity, electromagnetic force).

Example: Gravity pulls objects toward the Earth, while friction opposes motion between surfaces.

Newton's Laws of Motion

First Law (Law of Inertia)

An object at rest remains at rest, and an object in motion remains in motion at constant velocity unless acted upon by a net external force.

• Inertia: The tendency of an object to resist changes in its state of motion.

• Net Force: The vector sum of all forces acting on an object.

Example: A hockey puck slides on ice with little friction and continues moving until it hits something.

Second Law (Law of Acceleration)

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

Mathematical Formulation:

$\vec{F} = m \vec{a}$

• $\vec{F}$: Net force (in newtons, N)

• $m$: Mass (in kilograms, kg)

• $\vec{a}$: Acceleration (in meters per second squared, m/s2)

Example: If you push a cart, the harder you push (greater force), the faster it accelerates. A heavier cart (greater mass) accelerates less for the same force.

Third Law (Action-Reaction)

For every action, there is an equal and opposite reaction. Forces always occur in pairs.

• Action-Reaction Pairs: If object A exerts a force on object B, then object B exerts an equal and opposite force on object A.

Example: When you jump off a boat, you push the boat backward as you move forward.

Mass, Weight, and Density

Mass vs. Weight

Mass is a measure of the amount of matter in an object; it is a scalar quantity and does not change with location. Weight is the force of gravity acting on an object's mass; it is a vector and depends on the local gravitational field.

• Weight Formula: $w = mg$

• Units: Mass is measured in kilograms (kg); weight is measured in newtons (N).

Example: An object with a mass of 1 kg weighs 9.8 N on Earth but less on the Moon due to lower gravity.

Density

Density is the mass per unit volume of a substance. It is often represented by the Greek letter rho ($\rho$).

Density Formula: $\rho = \frac{m}{V}$

• Units: kg/m3 (SI), or lbs/ft3 (imperial)

Example: Gold is denser than silver; for the same volume, gold has more mass.

Material	Density (lbs)
Gold	1.3
Silver	0.7

Units and Measurement

Force Units

• Newton (N): The SI unit of force. $1\,\text{N} = 1\,\text{kg} \cdot \text{m}/\text{s}^2$

• Pound (lb): Commonly used in the United States. $1\,\text{lb} = 4.448\,\text{N}$

• Kip: 1 kip = 1000 lbs (used in engineering contexts)

Example: A force of 10 N is required to accelerate a 1 kg mass at 10 m/s2.

Measuring Force and Mass

• Spring Scale: Measures force by the extension of a spring, proportional to the force applied.

• Balance: Measures mass by comparing an unknown mass to a known mass.

Example: Weighing an object on a spring scale gives its weight (force), while a balance gives its mass.

Applications and Problem Solving

Calculating Acceleration and Force

To determine the net force or acceleration, use Newton's second law. For example, if a car slows from 100 km/h to rest over a certain distance, you can calculate the acceleration and then the net force required.

Example Calculation:

• Initial velocity: $v_0 = 100\,\text{km/h}$

• Final velocity: $v = 0$

• Acceleration: $a = -7.0\,\text{m/s}^2$

• Time and distance can be found using kinematic equations.

Volume and Its Limitations

Volume is not a reliable measure of mass because different materials have different densities, and volume can change with temperature and pressure.

Example: A block of gold and a block of silver with the same volume will have different masses due to their different densities.

Summary Table: Key Quantities and Units

Quantity	Symbol	SI Unit	Other Units
Force	F	Newton (N)	Pound (lb), Kip
Mass	m	Kilogram (kg)	Pound (lb)
Weight	w	Newton (N)	Pound (lb)
Density	\rho	kg/m3	lb/ft3

Additional info: Some context and examples were expanded for clarity and completeness, including the summary tables and explicit formulae.