Energy

Kinetic and Potential Energy (10.1)

Energy is a fundamental concept in physics, representing the capacity to do work. Two primary forms of mechanical energy are kinetic energy and potential energy.

• Kinetic Energy (K): The energy an object possesses due to its motion. It is given by the formula:

• where m is mass (kg) and v is speed (m/s).

• Unit: Joule (J), where .

• Example: A 2 kg object moving at 3 m/s has J.

• Gravitational Potential Energy (Ug): The energy an object has due to its position in a gravitational field, typically at height y above a reference point:

• where g is the acceleration due to gravity (9.8 m/s2).

• The value of depends on the choice of (reference level), but only changes in affect physical outcomes.

• Example: A 1 kg object at 5 m above ground: J.

Conservation of Mechanical Energy (10.3)

The mechanical energy of a system is the sum of its kinetic and potential energies:

• In the absence of non-conservative forces (like friction or air resistance), the total mechanical energy is conserved:

• Example: A ball dropped from height will have its initial potential energy fully converted to kinetic energy just before impact (neglecting air resistance).

Energy and Ratio Reasoning (10.3)

• Comparing Kinetic Energies: If two marbles, one twice as heavy as the other, are dropped from the same height, the heavier marble has twice as much kinetic energy just before hitting the ground, since (all potential energy converts to kinetic energy).

• Ramp Height and Speed: For a block sliding down a frictionless ramp, to double the speed at the bottom (), the ramp must be four times as high, since , so .

Lecture Problem Example

• Projectile from a Cliff: Conservation of energy can be used to find the speed of a projectile when it hits the ground:

• Initial kinetic and potential energies are set by launch speed and height; final kinetic energy is found when the projectile reaches ground level ().

Restoring Forces and Hooke's Law (10.4)

A restoring force is a force that acts to return a system to its equilibrium position. For springs, this is described by Hooke's Law:

• k is the spring constant (N/m), a measure of the spring's stiffness.

• is the displacement from equilibrium.

• The negative sign indicates the force is always directed opposite to the displacement.

• Example: If N/m and m, N.

Elastic Potential Energy (10.5)

The energy stored in a stretched or compressed spring is called elastic potential energy:

• This energy can be converted into kinetic energy when the spring is released.

• Example: Compressing a spring ( N/m) by 0.1 m stores J.

Energy and Ratio Reasoning (10.5)

• If a spring is compressed three times as far ( instead of ), the launch speed increases by a factor of , since and at release, so .

Lecture Problem Examples

• Block on Incline with Spring: Conservation of energy can be used to find the speed of a block after moving a certain distance down an incline, or how far it travels before stopping.

• Gymnast and Trampoline: The gymnast's kinetic and potential energy at takeoff and landing can be used to determine the speed at impact and the compression of the trampoline (modeled as a spring).

Energy Diagrams and Oscillation (10.6)

For systems like a mass on a spring, energy oscillates between kinetic and potential forms, but the total mechanical energy remains constant.

• Energy Diagram: Shows how potential energy (PE) and kinetic energy (KE) vary with position.

• At turning points, all energy is potential (); at equilibrium, potential energy is minimum and kinetic energy is maximum.

Energy Diagrams and Speed

• As potential energy decreases below total energy, kinetic energy increases and the object speeds up.

• As potential energy increases toward total energy, kinetic energy decreases and the object slows down.

Energy Diagrams and Equilibrium

• Stable Equilibrium: At a minimum of potential energy; if disturbed, the object oscillates about this point.

• Unstable Equilibrium: At a maximum of potential energy; if disturbed, the object moves away from this point.

Finding Force from Potential Energy

The force associated with a potential energy function is given by:

• The force at a point is the negative slope of the potential energy curve at that point.

Conservative and Nonconservative Forces (10.7)

Conservative forces are those for which the work done is independent of the path taken, and a potential energy function can be defined.

• For a conservative force:

• Examples: gravity, spring force.

• For nonconservative forces (e.g., friction), work depends on the path, and no potential energy function exists.

Example Potential Energy Calculations

• Gravity:

• Spring:

Thermal Energy and Energy Conservation (10.8)

When nonconservative forces like friction are present, some mechanical energy is transformed into thermal energy:

• fk is the kinetic friction force, is the distance slid.

The law of conservation of energy states that the total energy in a closed system remains constant:

• If no friction:

• If friction is present:

Lecture Problem Examples

• Cannonball in Sand: Use energy conservation to find the average friction force as the cannonball penetrates the sand.

• Puck on Inclined Ramp with Spring and Friction: Calculate the speed of the puck as it leaves the ramp, accounting for energy lost to friction.

Summary Table: Types of Forces and Energy Conservation

Additional info: These notes are based on standard college physics curriculum and expand on the provided slides for clarity and completeness.