Chapter 5: Applying Newton’s Laws

Overview

This chapter focuses on using Newton’s laws to solve equilibrium and dynamics problems. It covers the identification and analysis of forces, the use of free-body diagrams, and the application of Newton’s laws to a variety of physical situations, including objects at rest, moving at constant velocity, and accelerating.

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 sum of the forces in each direction (x and y) must be zero:

Example: An orangutan hangs from a rope. The tension in the rope equals the weight of the orangutan.

Solving Equilibrium Problems

• Strategize: Confirm the object is in equilibrium.

• Prepare: Identify all forces and draw a free-body diagram.

• Solve: Use Newton’s second law in component form to solve for unknown forces.

• Assess: Check units and reasonableness of the result.

Conceptual Example: Forces in Static Equilibrium

A rod on frictionless ice is lifted by a string. Only a vertical string orientation ensures equilibrium, as horizontal components would cause motion.

Dynamics and Newton’s Second Law

Newton’s Second Law

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

For component analysis:

Solving Dynamics Problems

• Strategize: Identify known forces or use kinematics to find acceleration.

• Prepare: Sketch diagrams, list knowns, and draw free-body diagrams.

• Solve: Use Newton’s second law and kinematic equations as needed.

• Assess: Check units and reasonableness.

Example: Putting a Golf Ball

A golf ball slows due to friction. Use Newton’s second law to find acceleration, then kinematics to determine if it reaches the hole.

Mass and Weight

Definitions

• Mass: A measure of an object’s inertia; constant regardless of location.

• Weight: The gravitational force exerted on an object by a planet; varies with location.

Weight is calculated as:

Apparent Weight

Your sensation of weight is due to contact forces supporting you. Apparent weight can differ from true weight when accelerating:

(normal force)

Normal Forces

Definition and Properties

The normal force is the perpendicular contact force exerted by a surface. It adjusts to keep objects from penetrating the surface.

Example: A book pressed down on a table experiences a normal force greater than its weight.

Normal Forces on an Incline

On an inclined plane, the normal force and weight can be decomposed into components:

Friction

Static Friction

Static friction prevents motion between surfaces. Its maximum value is:

• Direction opposes potential motion.

• Magnitude adjusts to prevent movement, up to .

Kinetic Friction

Kinetic friction acts when objects slide. Its magnitude is:

• Direction opposes motion.

• Magnitude is nearly constant, independent of speed.

Rolling Friction

Rolling friction occurs for wheels; it is generally less than kinetic friction and depends on the coefficient of rolling friction.

Causes of Friction

Friction arises from microscopic roughness and the interaction of surfaces.

Drag Forces

Definition and Reynolds Number

Drag opposes motion through a fluid and increases with speed. The Reynolds number determines whether inertial or viscous forces dominate:

• High Re: Inertial forces dominate; drag proportional to .

• Low Re: Viscous forces dominate; drag proportional to (Stokes' law).

Drag at High Reynolds Number

For most objects in air:

• is the drag coefficient.

• is cross-sectional area.

Terminal Speed

Terminal speed is reached when drag force equals weight:

Interacting Objects

Newton’s Third Law

Every force occurs as one member of an action/reaction pair. The two members act on different objects, are equal in magnitude, and opposite in direction.

Objects in Contact

When two objects are in contact, their accelerations are linked, and action/reaction pairs must be identified in separate free-body diagrams.

Ropes and Pulleys

Tension in Ropes

For massless ropes, tension is the same throughout and equals the force applied at the ends. Passing over a massless, frictionless pulley does not change the tension.

Summary Table: Types of Forces

Summary: Problem-Solving Strategies

• For equilibrium: ,

• For dynamics: Use Newton’s second law and kinematics

• Draw free-body diagrams for each object

• Identify action/reaction pairs for interacting objects

• For ropes and pulleys: Tension is constant in massless ropes and unaffected by ideal pulleys

Applications

• Apparent weight: Equals true weight only when vertical acceleration is zero.

• Terminal speed: Occurs when drag force balances weight.

• Strings and pulleys: Tension is constant and equal to the force applied.