Conservative and Non-Conservative Forces

Types of Forces

In physics, forces are classified as conservative or non-conservative based on their effect on the total mechanical energy of a system.

• Conservative Forces: These forces do not dissipate mechanical energy. The total mechanical energy (kinetic + potential) remains constant when only conservative forces act. Examples include gravity, spring force, and electrostatic force.

• Non-Conservative Forces: These forces cause the total mechanical energy to change, usually by converting mechanical energy into other forms such as heat. Examples include friction and drag force.

Spring Force and Elastic Potential Energy

Spring Force as a Conservative Force

A stretched or compressed spring stores energy as elastic potential energy. The spring force is a classic example of a conservative force.

• Elastic Potential Energy: The energy stored in a spring stretched or compressed by a distance x is given by:

• k is the spring constant (N/m).

• x is the displacement from the equilibrium position.

Example: Compressing a spring by 0.1 m with k = 500 N/m stores J of energy.

Conservation of Energy with Mass and Spring

Energy Conservation Principle

When a mass interacts with a spring, the total mechanical energy (kinetic + potential) is conserved if only conservative forces are present.

• Initial State: Spring relaxed, mass at rest.

• Final State: Spring compressed, mass may be moving.

The conservation of energy equation is:

For a mass m and spring constant k:

Example: If a 0.2 kg mass compresses a spring from 0.1 m to 0.08 m, you can solve for the final velocity using the above equation.

Conservation of Energy for an Object Falling onto a Spring

Total Mechanical Energy Conservation

When an object falls onto a spring, the total mechanical energy (kinetic, gravitational potential, and elastic potential) is conserved:

• K: Kinetic energy

• : Gravitational potential energy

• : Elastic potential energy

Example: A mass dropped from height d onto a spring will compress the spring until all energy is stored as elastic potential energy.

Non-Conservative Forces and Energy Dissipation

Effect of Non-Conservative Forces

When non-conservative forces (like friction) act, mechanical energy is not conserved. Instead, some energy is transformed into other forms (e.g., heat).

• Problems such as "How far does the box slide?" involve calculating energy lost to friction.

Example: A box slides down a frictionless incline and then across a rough surface. The distance it travels before stopping depends on the work done by friction.

Work Done by Non-Conservative Forces

Generalized Energy Equation

If non-conservative forces do work, the change in total mechanical energy equals the work done by these forces:

• is the work done by non-conservative forces (e.g., friction, drag).

Example: The energy lost to friction as a box slides across a rough surface can be calculated using this equation.

Power

Definition and Units

Power is a measure of how quickly work is done or energy is transferred.

• Mathematically, power is defined as:

• The SI unit of power is the watt (W), where .

• .

Example: If 100 J of work is done in 10 seconds, the power is W.

Average Power

When a quantity of work is done over a time interval , the average power is:

Power in Physical Situations

For example, if a runner does work against gravity, the average power output is:

where is mass, is acceleration due to gravity, and is the vertical displacement.

Applications and Problem Solving

Sample Problems

• Box on Inclined Plane: A box slides down a frictionless incline and then across a rough surface. Given the height and coefficient of kinetic friction, calculate how far the box travels before stopping.

• Roller Coaster Loop: A coaster car released from a height h travels along a frictionless track and enters a loop. Calculate the kinetic energy at the top of the loop, the minimum velocity to stay on the track, and the minimum initial height required.

• Slingshot and Hooke's Law: A slingshot obeying Hooke's law launches a pebble. Given the stretch distance and resulting height, use energy conservation to predict the height for different stretch distances.

Example Table: Energy Conservation Scenarios

Additional info: These notes cover topics from Chapter 5 (Applications of Newton's Laws) and Chapter 7 (Work & Energy) in a typical college physics curriculum, including the classification of forces, energy conservation, and power calculations.