Dynamics: Newton's Laws of Motion

Introduction to Dynamics

• Dynamics explains why bodies move the way they do, linking motion to forces.

• Example Questions:

  • Why does a dropped feather fall slower than a dropped baseball?

  • Why do you feel pushed backward when a car accelerates forward?

Newton's Laws of Motion

• Newton’s Laws of Motion were formulated by Sir Isaac Newton.

• First Law: An object at rest stays at rest, and an object in motion stays in motion unless acted upon by a net external force.

Second Law: An object accelerates when a net force is applied.

• Mathematical Formulation: $F_{net} = ma$ where:

  • Fnet​ is the net force

  • m is mass

  • a is acceleration

• Third Law: For every action, there is an equal and opposite reaction.

The Foundations of Classical Mechanics

• Newton's laws form the basis of classical (Newtonian) mechanics, applicable except in extreme conditions (near light speed or atomic level).

• Students may find Newton's laws counterintuitive, especially regarding common-sense ideas about forces and motion.

Learning Goals

1. Understand what force means in physics, why forces are vectors, and the significance of net force.

2. Comprehend the implications of a net force of zero and the concept of inertial frames of reference.

3. Apply the relationship between net force, mass, and acceleration.

4. Differentiate between mass and weight.

5. Analyze the interaction forces between two bodies.

6. Utilize free-body diagrams to analyze forces.

Common Types of Forces

Force: Definition and Magnitude

• A force is an interaction between two bodies or between a body and its environment.

• Forces are vector quantities (magnitude and direction).

  • Units: 1 Newton (N) is the standard unit of force in SI, defined precisely.

Types of Forces

• Contact Forces:

  • Normal Force: Acts perpendicular to the surface in contact.

  • Friction Force: Acts parallel to the contact surface, opposes motion.

  • Tension Force: Force transmitted through a string, rope, or cord.

• Long-Range Forces:

  • Examples include gravitational forces and magnetic interactions, which act at a distance without physical contact.

Superposition of Forces

• If two forces F₁ and F₂ act at the same point, their combined effect can be represented as a single resultant force $R = F₁ + F₂$.

  • $R$ represents the **resultant force**. It is the combined effect of two forces, $F₁$ and $F₂$, acting at the same point, found by vectorially adding them according to the principle of superposition of forces.

• The principle of superposition states that all forces acting on an object can be combined vectorially.

Newton’s First Law

• When the net force = 0:

  • If at rest, the object remains at rest.

  • If in motion, the object continues in motion with constant velocity and zero acceleration.

• Inertia: Resistance of an object to change its state of motion.

Inertial Frames of Reference

• A frame where Newton’s first law is valid is called an inertial frame.

• The Earth is approximately an inertial frame. Acceleration introduces non-inertial frames.

Newton’s Second Law

• Expressed mathematically as $F_{net} = ma$. It illustrates how net forces produce acceleration in an object.

• The direction of the force determines the direction of the acceleration.

• Applying a constant net force results in constant acceleration.

Mass and Force

• Inertial mass measures resistance to acceleration.

• $F_{net} = ma$ helps to understand acceleration in response to applied force.

• The SI unit for mass is the kilogram.

Forces and Motion Applications

• Example of a worker applying 20 N to a 40 kg box:

  • Acceleration $a = F/m = 20 N / 40 kg = 0.5 m/s²$

• A summary of force interactions is captured using free-body diagrams.

Newton's Third Law

• Forces occur in pairs; any force exerted by body A on body B results in an equal and opposite force by body B on body A.

• These forces act on different bodies, clarifying action-reaction pairs.

Free-Body Diagrams

• Essential for analyzing forces acting on an object.

• Illustrates which external forces are relevant to a body under consideration.

• Each force is represented as an arrow reflecting its direction and magnitude.

• Do not mix action-reaction forces on the same body in free-body diagrams.

Summary of Newton’s Laws of Motion

First Law:

 $R_{net} = 0 \implies \text{body remains at rest or moves with constant velocity}$ , represents Newton's First Law of Motion. It signifies that if the net force acting on a body is zero, the body's state of motion will not change; it will either remain at rest or continue to move with a constant velocity (zero acceleration). This is also known as the principle of inertia, which describes an object's resistance to changes in its state of motion.

Second Law:

• F=ma

• It states that the net force (the vector sum of all individual forces) acting on an object is directly proportional to the mass of the object and the acceleration it experiences. The acceleration

Third Law:

• FA on B=−FB on A
  This formula represents Newton's Third Law of Motion, stating that for every action, there is an equal and opposite reaction. It signifies that when body A exerts a force on body B (FA on BFA on B), body B simultaneously exerts a force on body A (FB on AFB on A) that is equal in magnitude but opposite in direction. Importantly, these action-reaction forces always act on different bodies.