Applying Newton’s Laws

Introduction

This chapter focuses on using Newton’s laws to analyze equilibrium and dynamics problems, including the effects of friction, drag, and interactions between objects. The concepts are illustrated with practical examples and problem-solving strategies.

Equilibrium

Static and Dynamic Equilibrium

Equilibrium occurs when the net force on an object is zero. There are two types:

• Static equilibrium: The object is at rest.

• Dynamic equilibrium: The object moves in a straight line at constant speed (a = 0).

In both cases, the sum of the forces in each direction is zero:

To solve equilibrium problems:

1. Check if the object is in equilibrium (is a = 0?).

2. Identify all forces and draw a free-body diagram.

3. Write Newton’s second law for each direction and solve for unknowns.

Example: Orangutan Hanging from a Rope

• The orangutan is at rest, so the tension in the rope equals its weight.

• Forces: upward tension T, downward weight w.

• Equilibrium:

Example: Equilibrium on a Frictionless Surface

• Forces must balance in both x and y directions.

• Only a vertical string (no horizontal component) can keep a rod at rest on frictionless ice.

Dynamics and Newton’s Second Law

Newton’s Second Law

Newton’s second law relates the net force on an object to its acceleration:

In component form:

To solve dynamics problems:

1. Identify all forces and draw a free-body diagram.

2. Write Newton’s second law in components.

3. Solve for acceleration or unknown forces as needed.

4. Use kinematics if required to find positions or velocities.

Example: Towing a Car at an Angle

• A car is towed at a steady speed by a rope at 20° above the horizontal, with friction opposing motion.

• Forces: tension T, friction f, normal force n, weight w.

• Equilibrium in x:

• Solve for tension:

Mass and Weight

Definitions

• Mass (m): A measure of an object’s inertia; does not change with location.

• Weight (w): The gravitational force on an object:

• Weight varies with gravitational field strength; mass does not.

Apparent Weight

• The force you feel (e.g., on a scale) is your apparent weight.

• When accelerating upward:

• When accelerating downward:

Weightlessness

• In free fall, apparent weight is zero, but true weight (gravitational force) remains.

Normal Forces

Definition and Properties

• The normal force is the perpendicular contact force exerted by a surface.

• It adjusts to balance other forces and prevent penetration of the surface.

Example: Book Pressed on a Table

• Forces: weight, normal force, and downward force from a hand.

• Normal force:

Normal Force on an Incline

• The normal force is always perpendicular to the surface.

• On an incline,

Friction

Static Friction

• Prevents relative motion between surfaces in contact.

• Adjusts up to a maximum value:

• Direction opposes impending motion.

Kinetic Friction

• Acts when surfaces slide past each other.

• Magnitude:

• Direction opposes motion; independent of speed.

Rolling Friction

• Occurs when an object rolls over a surface.

• Much smaller than kinetic friction:

Coefficients of Friction (Selected Values)

Drag Forces

Definition and Reynolds Number

• Drag force opposes motion through a fluid and increases with speed.

• The Reynolds number (Re) distinguishes between inertial and viscous drag:

• High Re (>1000): Inertial drag dominates; drag force

• Low Re (<1): Viscous drag dominates; drag force

High Reynolds Number Drag

• Drag force:

• is the drag coefficient, is cross-sectional area, is fluid density.

Terminal Speed

• Terminal speed is reached when drag force equals weight:

• At terminal speed, acceleration is zero.

Low Reynolds Number Drag (Stokes’ Law)

• For small, slow-moving objects:

• Applies to microscopic particles in fluids.

Interacting Objects, Ropes, and Pulleys

Newton’s Third Law

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

• Action/reaction pairs act on different objects.

Objects in Contact

• Draw separate free-body diagrams for each object.

• Write Newton’s second law for each object.

• Action/reaction forces have equal magnitude and opposite direction.

• Objects in contact have the same acceleration.

Ropes and Pulleys

• The tension in a massless rope is the same throughout its length.

• For a massless, frictionless pulley, tension is unchanged as the rope passes over it.

Summary Table: Key Forces and Equations

Problem-Solving Strategies

• Draw clear free-body diagrams for each object.

• Write Newton’s second law in component form.

• Identify all forces, including friction, drag, and tension.

• Check units and reasonableness of answers.