Work, Energy, and Conservation Principles

Gravitational Potential Energy

Gravitational potential energy is the energy an object possesses due to its position in a gravitational field, typically near the Earth's surface.

• Definition: The gravitational potential energy at height y is given by:

• Reference Point: The zero of potential energy is chosen at a convenient location, often at ground level or the lowest point in the problem.

• Change in Potential Energy: The change in gravitational potential energy as an object moves from to is:

Work-Energy Theorem and Conservation of Mechanical Energy

The work-energy theorem relates the change in kinetic energy to the net work done on a system. When only conservative forces (like gravity and springs) do work, mechanical energy is conserved.

• Work-Energy Theorem:

• Mechanical Energy: (kinetic plus potential energy)

• Conservation (no external work or non-conservative forces):

• With Dissipative Forces: If non-conservative forces (like friction) are present, their work is included as (thermal energy):

Example: Block Sliding Down an Inclined Plane with Friction

Problem: A mass is released from rest on an inclined plane (), , and travels down the slope. Find the final speed .

• System: Block + slope + Earth

• Energy Equation:

• Initial kinetic energy (starts from rest)

• Change in height:

• Friction work:

• Thermal energy:

Elastic Potential Energy

Elastic potential energy is stored in a spring or elastic object when it is compressed or stretched from its equilibrium position.

• Definition: For a spring with force constant and displacement :

• Change in Elastic Potential Energy:

• Work Done by Spring: The work done by the spring on an object is equal to the negative change in the spring's potential energy.

Example: Spring Launch on an Inclined Ramp

Problem: A spring () is compressed by . A mass is released from rest, travels up a ramp at . How far from the ramp does the mass land? (No friction.)

• Strategy: Find launch speed using energy conservation, then use projectile motion to find .

• Energy Conservation:

• Initial: , ,

• Final: , , (where )

• Projectile Motion: Use to find horizontal distance .

• Vertical motion:

• Horizontal motion:

• Solve for when (ground level):

• Horizontal distance:

Classic Energy Conservation Problem: Roller Coaster Loop

Problem: What is the minimum height required for a roller coaster to complete a vertical loop of radius without losing contact at the top? (No friction.)

• Key Principle: At the top of the loop, the normal force must be at least zero for the car to stay on the track. This requires:

• Energy Conservation: Set initial potential energy at height equal to the sum of potential and kinetic energy at the top of the loop:

• Substitute from the force condition:

• Minimum Height: The car must start from at least times the loop radius above the bottom to complete the loop safely.

Summary Table: Types of Potential Energy

Additional info: These notes cover key concepts from Ch 10 (Interactions and Potential Energy) and Ch 09 (Work and Kinetic Energy), with applications to Ch 11 (Impulse and Momentum) and Ch 04 (Kinematics in Two Dimensions).