Applying Newton’s Laws

Introduction

Newton’s laws of motion provide the foundation for analyzing the forces acting on objects and predicting their motion. This chapter focuses on applying these laws to solve problems involving equilibrium, dynamics, friction, and circular motion.

Equilibrium and Newton’s First Law

Conditions for Equilibrium

An object is in equilibrium if it is at rest or moving with constant velocity in an inertial frame of reference. According to Newton’s first law, the net force on a body in equilibrium must be zero:

• Translational Equilibrium: The sum of all forces acting on the object is zero.

• Component Form: The sum of the x-components and y-components of all forces must each be zero.

Problem-Solving Strategy for Equilibrium

• Draw a sketch of the physical situation.

• Draw a free-body diagram for each body in equilibrium, showing all forces acting on it.

• Identify all forces (contact and non-contact) and use for weight if mass is given.

• Choose coordinate axes and resolve forces into components.

• Set up equations for the sum of forces in each direction and solve for unknowns.

Dynamics and Newton’s Second Law

Newton’s Second Law

When the net force on a body is not zero, the body accelerates in the direction of the net force. Newton’s second law relates the net force to the acceleration:

Problem-Solving Strategy for Dynamics

• Draw a sketch and a free-body diagram for each moving body, labeling all forces (e.g., weight ).

• Choose coordinate axes and resolve forces into components.

• Write Newton’s second law for each direction and each body.

• List knowns and unknowns, identify target variables, and solve the equations.

Free-Body Diagrams: Correct and Incorrect Practices

• Only actual forces (e.g., gravity, tension, normal force, friction) should appear in a free-body diagram.

• The acceleration vector can be drawn to the side for reference, but is not a force and should not be included as a force vector.

Frictional Forces

Nature of Friction

Friction is a force that opposes the relative motion or tendency of such motion of two surfaces in contact. It arises from molecular interactions at the contact surfaces.

• Kinetic friction (): Acts when a body slides over a surface.

• Static friction (): Acts when there is no relative motion.

Static and Kinetic Friction: Sequence of Events

• No applied force:

• Weak applied force:

• Stronger applied force: (object about to move)

• Object slides: (kinetic friction)

Coefficients of Friction

The coefficients of static and kinetic friction depend on the materials in contact. Typical values are shown below:

Applications: Stick-Slip Phenomena

Stick-slip motion, such as the squeak of windshield wipers on dry glass, occurs when static friction alternates with kinetic friction as the wiper blade sticks and then slips.

Fluid Resistance and Terminal Speed

Drag Force and Terminal Velocity

When an object moves through a fluid (like air), it experiences a resistive force (drag) that increases with speed. Eventually, the drag force balances the weight, and the object reaches terminal speed:

• Before terminal speed: or

• At terminal speed : or

Dynamics of Circular Motion

Uniform Circular Motion

For a particle in uniform circular motion, the net force and acceleration are always directed toward the center of the circle (centripetal direction):

• Velocity is tangent to the circle.

• Acceleration and net force point toward the center.

What Happens if the Centripetal Force Disappears?

If the force maintaining circular motion (e.g., a string) breaks, the object moves in a straight line tangent to the circle, obeying Newton’s first law.

Common Error: The 'Centrifugal Force'

In an inertial frame, there is no real outward 'centrifugal force.' The correct free-body diagram includes only real forces; the acceleration vector can be shown to the side for clarity, but is not a force and should not be drawn as one.

The Fundamental Forces of Nature

Overview of Fundamental Interactions

All forces in nature are manifestations of four fundamental interactions:

• Gravitational

• Electromagnetic

• Strong nuclear

• Weak nuclear

Physicists aim to unify these interactions into a comprehensive 'theory of everything.'