Chapter 5: Applying Newton’s Laws

Introduction

This chapter focuses on the application of Newton’s laws of motion to solve problems involving equilibrium, dynamics, and friction. It provides systematic approaches for analyzing forces, constructing free-body diagrams, and solving for unknowns in both static and dynamic situations.

Equilibrium

Static Equilibrium

An object is in static equilibrium when it is at rest and the net force acting on it is zero.

• Definition: Static equilibrium occurs when an object remains at rest, with no net force acting upon it.

• Mathematical Condition:

• Example: A book resting on a table is in static equilibrium because the upward normal force balances the downward gravitational force.

Dynamic Equilibrium

Dynamic equilibrium refers to an object moving at a constant velocity (no acceleration) in a straight line, with the net force still zero.

• Definition: An object in motion with constant speed and direction (no acceleration, ).

• Mathematical Condition:

• Example: A hockey puck sliding across frictionless ice at constant speed.

Equilibrium Problem-Solving Approach

To solve equilibrium problems, follow these steps:

1. Draw a diagram of the situation.

2. Identify all forces acting on the object.

3. Draw a free-body diagram to visualize forces.

4. Determine known and unknown quantities.

5. Write equations from Newton’s second law:

6. Solve for the unknowns using the equations.

Dynamics

Dynamics Problem-Solving Approach

Dynamics problems involve objects with acceleration. Newton’s second law connects forces and kinematics.

1. Draw a diagram of the scenario.

2. Identify all forces acting on the object.

3. Draw a free-body diagram.

4. Determine known and unknown quantities.

5. Write equations from Newton’s second law:

6. Use kinematics equations to find acceleration if needed.

7. Solve for the unknowns.

• Example: Calculating the tension in a rope pulling a car with friction and acceleration.

Worked Example Problems

Example 5.1: Car Towed at Constant Speed

• Given: Car mass = 1500 kg, rope at 20° above horizontal, friction force = 320 N.

• Find: Tension in the rope.

• Approach: Use equilibrium (no acceleration), resolve forces, and solve for tension.

Example 5.2: Car Accelerating

• Given: Same car, friction force = 320 N, car accelerates from rest to 12 m/s in 10 s.

• Find: Tension in the rope during acceleration.

• Approach: Use dynamics, calculate acceleration, apply Newton’s second law, and solve for tension.

Key Concepts

Mass and Weight

• Mass (m): The amount of matter in an object, measured in kilograms (kg). Mass is constant everywhere in the universe.

• Weight (W): The gravitational force on an object, measured in newtons (N). Weight depends on local gravity.

• Example: An object with mass 10 kg has weight N on Earth.

Apparent Weight

• Definition: The normal force (supporting force) felt by an object, which may differ from true weight if the object is accelerating.

• Elevator Accelerating Upward:

• Elevator Accelerating Downward:

• Weightlessness: Occurs when the supporting force is zero (e.g., in free fall).

Normal Force

• Definition: The perpendicular supporting force exerted by a surface on an object resting on it.

• On a horizontal surface:

• On an inclined plane:

Friction

Static Friction

• Definition: The force that resists the initiation of sliding motion between two surfaces.

• Maximum static friction:

• = coefficient of static friction

• Example: A box at rest on a table resists being pushed until the applied force exceeds .

Kinetic Friction

• Definition: The force that opposes the motion of two surfaces sliding past each other.

• Magnitude:

• = coefficient of kinetic friction

• Example: A sled sliding on snow experiences kinetic friction.

Rolling Friction

• Definition: The resistance force that slows down the motion of a rolling object.

• Magnitude:

• = coefficient of rolling friction

Coefficients of Friction (Table)

The following table summarizes typical coefficients of friction for various material pairs:

Drag Forces and Terminal Speed

Drag Force

• Definition: The resistive force exerted by a fluid (air or liquid) on an object moving through it, always opposite to the velocity.

• Magnitude: Increases with speed.

Reynolds Number

• Definition: A dimensionless number that predicts flow patterns in different fluid flow situations.

• = fluid density, = velocity, = characteristic length, = viscosity

Drag Force Equations

• High Reynolds number (turbulent flow):

• = drag coefficient, = cross-sectional area

• Low Reynolds number (laminar flow):

• = object radius

Terminal Speed

• Definition: The constant speed reached by an object when the drag force equals the gravitational force, resulting in zero net acceleration.

• Condition:

• Example: A skydiver reaches terminal speed when air resistance balances weight.

Objects in Contact and Tension

Solving Objects in Contact

• Draw a separate free-body diagram for each object.

• Write Newton’s second law for each object.

• For action-reaction pairs, equate the magnitudes:

• Objects in contact have the same acceleration.

Tension in Strings and Pulleys

• Massless String: Tension is the same throughout the string.

• Frictionless Pulley: Tension remains unchanged on both sides of the pulley.

• Example: Two masses connected by a string over a pulley experience equal tension if the pulley is massless and frictionless.

Summary Table: Problem-Solving Steps

Additional info: Some context and equations were inferred and expanded for completeness and clarity, based on standard college physics curriculum.