Conservation of Energy

Principle of Conservation of Energy

The conservation of energy is a fundamental concept in physics stating that the total mechanical energy of a system remains constant if only conservative forces are acting (assuming no work is done by external forces and no energy is lost to friction or other dissipative forces).

• Mathematical Statement:

• : Kinetic energy

• : Potential energy (gravitational, elastic, etc.)

• : Change in thermal (dissipative) energy, e.g., due to friction

If friction is the only dissipative force present:

Types of Potential Energy

• Gravitational Potential Energy: (where is the height above a reference point)

• Elastic (Spring) Potential Energy: (where is the spring constant and is the displacement from equilibrium)

Applications of Energy Conservation

Classic Energy Conservation Problem: Roller Coaster Loop

Consider a roller coaster car of mass starting from rest at height and entering a vertical loop of radius . The goal is to determine the minimum height required for the car to complete the loop safely (without losing contact at the top).

• System: Car, track, Earth

• Assumptions: No friction (), car starts from rest

Energy Conservation Equation:

• is the speed at the top of the loop

• At the top, the minimum speed is required to maintain contact (normal force )

Force Analysis at the Top:

Substitute into the energy equation:

• Result: The minimum height required is

Example: Block Sliding Off a Frictionless Globe

A small block of mass starts from rest at the top of a large frictionless globe of radius . The block slides down and leaves the surface at a certain height above the bottom. Find $h_{exit}$.

• System: Earth, globe, block

• Initial energy: (top of globe)

• Final energy: ,

Energy Conservation:

Force Analysis (when block leaves the globe):

Substitute into the energy equation:

From the geometry of the globe:

• Result: The block leaves the globe at a height above the bottom.

Energy Diagrams

Definition and Use

An energy diagram is a graph showing a system's potential energy and total energy as a function of position. It is a useful tool for visualizing how energy is distributed and conserved in a system.

• If , then

Mass-Spring System

Energy in a Mass-Spring System

For a mass attached to a spring of constant , the total mechanical energy is the sum of kinetic and elastic potential energy:

• Potential Energy Curve: The elastic potential energy forms a parabola as a function of displacement from equilibrium.

• Turning Points: The points where all energy is potential () correspond to the maximum displacement (, ).

• Oscillation: The mass oscillates between these turning points, converting energy between kinetic and potential forms.

Summary Table: Types of Potential Energy

Additional info: These notes cover material relevant to Chapter 9 (Work and Kinetic Energy) and Chapter 10 (Interactions and Potential Energy), as well as applications from Chapter 5 (Force and Motion) and Chapter 6 (Dynamics I: Motion Along a Line).