Conservation Laws

Energy and Conservation Principles

Conservation laws are fundamental in physics, describing quantities that remain unchanged in isolated systems. The mechanical energy of a system changes only due to work done by non-conservative forces.

• Mechanical Energy Conservation: , where is work by non-conservative forces.

• Conserved Quantities: Energy, momentum, and angular momentum are key examples.

• Examples: Free fall, frictionless roller coasters, and pendulums exhibit energy conservation.

Linear Momentum

Definition and Properties

Linear momentum is a vector quantity defined as the product of an object's mass and velocity. It is a central concept in analyzing motion and collisions.

• Formula:

• SI Unit: kg·m/s

• Vector Nature: Direction is the same as velocity.

Newton's Second Law and Momentum

Newton's second law relates the change in momentum to the net external force acting on a system.

• If mass is constant:

• Newton's 2nd Law (in terms of momentum):

Impulse

Definition and Relationship to Momentum

Impulse is the effect of a force applied over a time interval, resulting in a change in momentum.

• Formula:

• Change in Momentum:

• Direction: Impulse has the same direction as the average force.

Example: Golf Ball Bouncing Off Floor

A 50-gram golf ball strikes a hard floor at 30° with a speed of 40 m/s before and after impact. The impulse applied by the floor is calculated by resolving velocity components and applying the impulse-momentum theorem.

• Magnitude:

• Result: N·s

Applications and Conceptual Questions

Kinetic Energy and Momentum

Relationship between kinetic energy and momentum for a particle of mass :

• If is quadrupled, is doubled:

Momentum Change in Collisions

Comparing momentum change for balls dropped from the same height:

• Case 1 (bounces): Greater momentum change due to reversal of velocity.

• Case 2 (sticks): Smaller momentum change.

Impulse and Force in Collisions

Directing a car into a haystack vs. a stone wall:

• Haystack: Smaller average force due to longer collision time.

• Impulse:

Total Linear Momentum of a System

Definition

The total linear momentum of a system is the vector sum of the momenta of all objects in the system.

• For two objects:

Example: Two Robins in Flight

Calculate the total momentum by resolving each bird's velocity into components and summing vectorially.

Isolated Systems

Internal vs. External Forces

Internal forces act between parts of the system; external forces are exerted by agents outside the system.

• Isolated System: Sum of external forces is zero.

• Momentum is conserved in isolated systems.

Conservation of Linear Momentum

Principle

If the net external force on a system is zero, the total linear momentum remains constant.

• If , then

• Conservation:

Example: Ice Skaters

Two skaters push off each other on frictionless ice. Use conservation of momentum to find the velocity of one skater given the other's velocity and both masses.

Collisions

Types of Collisions

Collisions are events where momentum and energy are exchanged. Total linear momentum is conserved in isolated systems.

• Elastic Collision: Total kinetic energy and momentum are conserved. ,

• Inelastic Collision: Momentum is conserved, but kinetic energy is not. Some energy is lost as heat or deformation.

• Completely Inelastic Collision: Objects stick together after collision.

Mathematical Treatment

Example: Colliding Projectiles

Calculate final velocities using conservation of momentum for objects moving in different directions. Use vector components and magnitude calculation:

Circular Motion

Introduction

Circular motion involves objects moving along a circular path, described by position, velocity, acceleration, and force. Newton's laws apply to these scenarios.

• Objects in circular motion experience acceleration even at constant speed.

• Two types: Revolution (orbiting a focal point) and Rotation (spinning on an axis).

Tangential Velocity

The instantaneous velocity at any point in circular motion is tangent to the path.

• For a full revolution: , where is radius and is period.

Angular Velocity

Angular velocity measures the rate of change of angular position.

• For a full revolution:

• Period:

Converting Between Linear and Rotational Quantities

Centripetal Forces

Definition and Direction

Centripetal force is the net force required to keep an object moving in a circular path, always directed toward the center.

• Formula:

• Also:

• It is a 'center-seeking' force.

Conceptual Example

When a ball on a string is swung in a circle and the string breaks, the ball moves tangentially to the circle at the point of release.

Rotational Kinematics and Angular Momentum

Angular Momentum

Angular momentum is the rotational analog of linear momentum, defined for rotating objects.

• Formula: , where is moment of inertia and is angular velocity.

• Conservation: In the absence of external torque, angular momentum is conserved.

Example: Figure Skater

A skater spinning with arms extended pulls them in, reducing moment of inertia and increasing angular speed to conserve angular momentum.

Additional info: These notes cover topics from Ch 08 (Linear Momentum and Collisions), Ch 10 (Rotational Motion and Angular Momentum), and Ch 06 (Uniform Circular Motion and Gravity) of a college physics course.