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 foundational for understanding how forces govern motion in everyday and engineered systems.

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.

In both cases, the acceleration is zero: .

For equilibrium in two dimensions:

All forces must be identified and represented in a free-body diagram.

Example: Orangutan Hanging from a Rope

An orangutan weighing 500 N hangs at rest from a vertical rope. The tension in the rope equals the weight:

Conceptual Example: Rod on Frictionless Ice

A rod is lifted by a string on frictionless ice. Only when the string is vertical (case b) can the net force be zero, since frictionless ice cannot exert a horizontal force.

Equilibrium with Multiple Forces: Chandelier Example

For a chandelier supported by cords at angles, resolve forces into components and apply equilibrium conditions:

Dynamics and Newton’s Second Law

Newton’s Second Law

Newton’s second law relates net force to acceleration:

To solve dynamics problems:

1. Identify all forces (draw a free-body diagram).

2. Write Newton’s second law in component form.

3. Solve for unknowns (acceleration, force, etc.).

Example: Towing a Car

A 1500 kg car is towed at constant speed by a rope at 20° above the horizontal. Friction opposes with 320 N. The tension is found by resolving forces and applying equilibrium:

Mass and Weight

Definitions

• Mass (m): A measure of an object’s inertia (resistance to acceleration).

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

Weight varies with location (e.g., different planets), but mass does not.

Apparent Weight

Your sensation of weight is due to the normal force supporting you. Apparent weight can differ from true weight if you are accelerating:

• When accelerating upward:

• When accelerating downward:

Normal Forces

Definition and Calculation

The normal force is the perpendicular contact force exerted by a surface. It adjusts to prevent penetration of the surface. On a horizontal surface:

• (if no other vertical forces)

If additional downward force is applied, .

Normal Force on an Incline

On an inclined plane, the normal force is:

Friction

Static Friction

Static friction prevents relative motion up to a maximum value:

Where is the coefficient of static friction. The direction opposes impending motion.

Kinetic Friction

Kinetic friction acts when objects slide:

Where is the coefficient of kinetic friction. It is usually less than and does not depend on speed.

Drag Forces

High Reynolds Number (Inertial Drag)

For large, fast-moving objects in fluids (e.g., cars, balls):

Where is the drag coefficient, is fluid density, is cross-sectional area, and is speed.

Low Reynolds Number (Viscous Drag)

For small, slow-moving objects (e.g., pollen in air):

• Stokes’ Law:

Where is viscosity, is radius, is speed.

Terminal Speed

Terminal speed is reached when drag force equals weight, resulting in zero acceleration:

• Set and solve for .

Interacting Objects, Ropes, and Pulleys

Newton’s Third Law

Every force is part of an action/reaction pair acting on different objects, equal in magnitude and opposite in direction.

Objects in Contact

When two objects are in contact and move together, analyze each with a separate free-body diagram. Their accelerations are equal if they move together.

Ropes and Pulleys

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

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

Summary Table: Common Forces and Equations

Key Problem-Solving Steps

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

2. Choose a coordinate system and resolve forces into components.

3. Apply Newton’s laws in component form.

4. Solve for unknowns (forces, acceleration, etc.).

5. Check units and reasonableness of your answer.