Chapter 5: Using Newton's Laws – Friction, Circular Motion, Drag Forces

Overview

This chapter explores the application of Newton's Laws to systems involving friction, circular motion, and drag forces. These concepts are fundamental in understanding the dynamics of objects in real-world scenarios, such as vehicles on roads, objects moving through fluids, and bodies in rotational motion.

5-1 Using Newton's Laws with Friction

Introduction to Friction

Friction is a resistive force that arises when an object moves on a surface or through a medium. It is due to interactions between the object and its environment and always acts opposite to the direction of motion.

• Frictional Force: The force that opposes relative motion between two surfaces in contact.

• Nature of Friction: Primarily arises from electromagnetic interactions at the microscopic level between atoms and molecules of the surfaces.

• Types of Friction: Static friction (prevents motion) and kinetic friction (opposes ongoing motion).

Example: A block sliding on a table experiences friction that slows it down and eventually stops it.

5-1-1 The Force of Kinetic Friction

Properties of Kinetic Friction

Kinetic friction acts when two surfaces are sliding past each other. Its magnitude is generally less than that of static friction.

• Proportionality: The force of kinetic friction is proportional to the normal force.

• Direction: Always opposite to the direction of motion.

• Coefficient of Friction: The proportionality constant depends on the nature of the surfaces in contact.

Formula:

where is the kinetic friction force, is the coefficient of kinetic friction, and is the normal force.

Static Friction Forces

Understanding Static Friction

Static friction prevents relative motion between two surfaces at rest with respect to each other. It adjusts up to a maximum value to prevent slipping.

• Variable Magnitude: Increases with applied force up to a maximum.

• Maximum Static Friction:

• Impending Motion: The equality sign is used only when the object is on the verge of moving.

Formula:

where is the static friction force, is the coefficient of static friction, and is the normal force.

Example: A book resting on a table does not move until the applied force exceeds the maximum static friction.

Comparing Static and Kinetic Friction

Static friction is generally greater than kinetic friction for the same pair of surfaces. This means it is easier to keep an object sliding than to start its motion.

• Static Friction: Prevents motion up to a threshold.

• Kinetic Friction: Opposes motion once sliding begins.

Coefficients of Friction

Typical Values

The coefficients of friction depend on the materials in contact. They are nearly independent of the area of contact and, for most practical purposes, are treated as constants.

5-2 Uniform Circular Motion – Kinematics

Motion in a Circle

Uniform circular motion refers to motion along a circular path at constant speed. The instantaneous velocity is always tangent to the circle.

• Radial (Centripetal) Acceleration: Points toward the center of the circle.

• Formula:

• Velocity: , where is the period.

Example: A ball revolving in a circle experiences centripetal acceleration directed inward.

5-3 Dynamics of Uniform Circular Motion

Forces in Circular Motion

For an object to move in a circle, a net force must act toward the center (centripetal force). This force can be provided by tension, friction, gravity, or other means depending on the scenario.

• Centripetal Force:

• No Centrifugal Force: There is no real outward force; the tendency to move straight is due to inertia.

Example: A ball on a string, a car on a curve, or a particle in a centrifuge all require inward force to maintain circular motion.

5-4 Highway Curves: Banked and Unbanked

Unbanked Curves

On a flat curve, friction between the tires and the road provides the centripetal force needed for a car to turn.

• Risk of Skidding: If friction is insufficient, the car will skid outward.

• Static vs. Kinetic Friction: Static friction is preferable; kinetic friction is lower and less controllable.

Banked Curves

Banking the road can help supply the required centripetal force through the normal force, reducing reliance on friction.

• Banking Angle Formula:

• Design Speed: For a given radius and speed, the banking angle can be calculated so that no friction is required.

Example: Expressway off-ramp design uses banking to allow safe turns at specific speeds.

5-5 Nonuniform Circular Motion

Acceleration Components

When speed varies along a circular path, the object has both radial (centripetal) and tangential acceleration components.

• Radial Acceleration:

• Tangential Acceleration:

Application: This analysis applies to any curved path, not just perfect circles.

5-6 Velocity-Dependent Forces: Drag and Terminal Velocity

Drag Forces

Objects moving through fluids experience drag forces that depend on their velocity.

• Low Speeds: Drag force is proportional to velocity ().

• High Speeds: Drag force is proportional to the square of velocity ().

Terminal Velocity

When the drag force equals the gravitational force, the object reaches a constant speed called terminal velocity.

• Condition:

• Terminal Velocity Formula (linear drag): , where is the drag coefficient.

Example: A skydiver eventually stops accelerating and falls at terminal velocity.

Glossary

• Friction: The resistive force opposing motion between surfaces.

• Frictional Force: The specific force due to friction.

• Terminal Velocity: The constant speed reached when drag balances gravity.