Conservation of Energy

Law of Conservation of Energy

The Law of Conservation of Energy states that the total energy of an isolated system remains constant, even as energy transforms from one form to another. This principle is fundamental in physics and applies to all physical processes where energy is neither created nor destroyed.

• Isolated System: A system where no energy is transferred into or out of the system (i.e., no external work is done).

• Energy Transformation: Energy can change forms (e.g., from potential to kinetic), but the total amount remains unchanged.

• Mechanical Energy: The sum of kinetic and potential energies in a system. In the absence of non-conservative forces (like friction), mechanical energy is conserved.

Mathematical Statement:

Where is the work done on the system. For an isolated system ():

Thus, the total energy remains constant.

Work-Energy Theorem

Statement and Application

The work-energy theorem relates the net work done on a system to the change in its kinetic energy. It is a powerful tool for solving problems involving energy transfers.

• Work-Energy Equation: The change in total energy of a system equals the work transferred to or from the system.

• Potential Energy: The theorem applies to potential energy as well as kinetic energy.

Equation:

Or, for mechanical energy (kinetic + potential):

Where is kinetic energy and is potential energy.

Types of Mechanical Energy

Kinetic and Potential Energy

• Kinetic Energy (): The energy of motion.

• Gravitational Potential Energy (): The energy due to position in a gravitational field.

• Elastic Potential Energy (): The energy stored in a stretched or compressed spring.

Solving Conservation of Energy Problems

General Approach

• Identify the system and determine if it is isolated (no external work) or if work is done on/by the system.

• Write the energy conservation equation, including all relevant forms of energy and work.

• Substitute known values and solve for the unknown quantity (e.g., velocity, height, displacement).

• Note: If only kinetic and gravitational potential energy are involved and the system is isolated, mass often cancels out when solving for velocity or height.

Worked Examples

Example 1: Car vs. Bicycle Down an Incline

Problem: A 1000 kg car and an 85 kg bicycle (with rider) are at rest at the top of a 100 m incline. Both are released simultaneously. Neglect friction. Who reaches the bottom with greater velocity?

• Known: kg, kg, m,

• Find: Final velocity at the bottom for each.

Solution:

Energy conservation (no work done, frictionless):

Solving for :

Mass cancels out, so both objects reach the bottom with the same velocity, regardless of mass.

Conclusion: Both the car and the bike reach the ground with the same speed.

Example 2: Water Slide

Problem: Katie starts from a height of 9.0 m with an initial speed of 2.0 m/s on a frictionless slide. What is her speed at the bottom?

• Known: m, m/s, m

• Find: Final velocity at the bottom.

Solution:

Energy conservation (no work done, frictionless):

Solving for :

Substitute values ( m/s):

m/s

Example 3: Trailer Pulled Up a Slope

Problem: Monica pulls a 25 kg trailer (with Jessie) up a 100 m slope rising 4.0 m. The bike exerts a constant force of 8.0 N. Initial speed is 5.3 m/s. What is the speed at the top?

• Known: kg, m, m, N, m/s

• Find: Final velocity at the top.

Solution:

Work is done on the system (trailer + Jessie):

Set (initial height = 0):

Solving for :

Substitute values:

m/s

Conclusion: The trailer's speed decreases as it ascends the slope, since the work done is not enough to fully counteract gravity.

Summary Table: Key Equations

Additional info: In real-world scenarios, friction and thermal energy may play a role, but these are often neglected in introductory problems for simplicity. The principles outlined here form the foundation for more advanced studies in energy conservation and dynamics.