Back to Energy

**Table of Contents**

- Work
- Forms of Energy
- The Two Basic Forms of Energy
- Potential and Kinetic Energy of the Pendulum
- Work and Heat: Forms of Energy Transfer, Not Forms of Energy
- Law of Conservation of Energy, Emmy Noether’s Proof
- Conservation of Energy in Classical and Quantum Mechanics

**Work**

- Energy is the capacity for doing work.
- Work is the energy transferred as a force moves an object
- britannica.com/science/work-physics
- “Work is the measure of energy transfer that occurs when an object is moved over a distance by an external force. If the force is constant, work may be computed by multiplying the length of the path by the magnitude of the force.”
- “Work done on a body is equal to the increase in the energy of the body, for work transfers energy to the body.}
- “Work and energy are expressed in the same units, for example, joules and foot-pounds.”

- Formal Definition
- Work by a force on an object over a distance
*the force exerted times the distance*covered. - For a constant force in a straight line
*W = F x D*

- For a variable force in a straight line from a to b

- Work by a force on an object over a distance
- Dropping a bowling ball
- If a sixteen pound bowling ball is dropped five feet, Earth’s gravity does work on the ball equalling:
- 16 pounds of force x 5 feet travelled = 80 foot-pounds of work.

- If a sixteen pound bowling ball is dropped five feet, Earth’s gravity does work on the ball equalling:
- The cue ball, with 10.6 joules of kinetic energy, hits a stationary eight ball:
- Before impact:
- Cue ball has kinetic energy of 10.6 joules
- Eight ball has zero.

- At Impact:
- 10.6 joules of work is performed, transferring 10.6 joules of energy from the cue ball to the eight ball

- After impact:
- The cue ball has zero kinetic energy
- Eight ball has kinetic energy of 10.6 joules

- Before impact:
- The formula
is used to derive formulas for different forms of energy*W = F x D*- To derive the formula for kinetic energy, for example, you calculate the amount of work required to move a mass
from rest to*m*meters per second.*v* - That amount of work is
**½**joules.*mv*^{2}

- To derive the formula for kinetic energy, for example, you calculate the amount of work required to move a mass

**Forms of Energy**

**Forms of Energy**

###### The **Forms**

- Total energy
- The total energy of a system is the sum of the forms of energy in the system, each calculated by its formula.

- Forms of energy include
- Chemical energy
- Elastic potential energy
- Electrical energy
- Gravitational potential energy
- Kinetic energy
- Mass energy
- Nuclear energy
- Radiant energy
- Thermal energy

###### Their **Formulas**

**Formulas**

### The **Two Basic Forms** of Energy

**Two Basic Forms**- Kinetic Energy, the energy of motion
- The kinetic energy of an object moving at a given speed is the work required to accelerate the object from rest to that speed.
*KE = ½ mv*is derived from^{2}and*W = FD**F = MA*

- Potential Energy, the energy of location (in a force field)
- The potential energy of an object at a given location is the work required to move the object from a reference point to that location.
- For example,
- is derived from
*W = FD*

- For example,
- Force and potential energy are intimately connected:
- Force = change of potential energy divided by distance

- For example, electric forces among atoms bend water molecules
- In the diagram the forces push downhill, from higher potential energy to lower.

- The potential energy of an object at a given location is the work required to move the object from a reference point to that location.

### Potential and Kinetic Energy of the **Pendulum**

**Pendulum**- The total energy of a pendulum system at a time = the kinetic energy of the bob at the time + the gravitational potential energy of the system at the time.
- The kinetic energy of the bob, calculated by
**½**, changes from zero to the total energy as the bob goes from its high point to its low point*mv*^{2}= mass of bob*m*= velocity of bob*v*

- The gravitational potential energy, calculated by
, changes from the total energy to zero as the bob goes from its high point to its low point*gmh*= acceleration due to gravity*g*= mass of bob*m*

= height of bob*h*

**Work and Heat**:

**Forms of Energy Transfer**,

**Not Forms of Energy**

**Work and Heat**- Work is a form of energy transfer, not a form of energy
*Work*is the energy transferred resulting from a force moving an object

- Heat is a form of energy transfer, not a form of energy
*Heat*is the energy transferred resulting from a difference in temperature

- The total energy of a system does not include work and heat.
- Water Bucket Analogy
- A first bucket has five quarts of water and a second two. You pour one quart from the first bucket into the second. The total amount of water, seven quarts, remains the same but it’s now divided differently: 4 + 3 vs 5 + 2.
- The seven quarts
**does not**include the one quart transferred.

**Law of Conservatio**n of Energy,

**Emmy Noether’s Proof**

- In 1915 Emmy Noether (EM-mee ΝEW-tar) proved
*Noether’s Theorem,*which says:- If the laws of physics remain valid through transformation X, then quantity Y is conserved, where
**X = translations in time and Y = energy**- X = translations in space and Y = linear momentum
- X = rotations in space and Y = angular momentum
- X = gauge transformations and Y = electric charge

- If the laws of physics remain valid through transformation X, then quantity Y is conserved, where
- Thus, Noether proved that the Law of the Conservation of Energy follows from the fact that the laws of physics remain valid through time.

**Conservation of Energy in Classical and Quantum Mechanics**

**Conservation of Energy in Classical and Quantum Mechanics**

**Proof of Conservation of Energy in Classical Mechanics***E = ½ mv*^{2}*+ V(x)*- The total energy of a system = kinetic energy + potential energy

- [several steps of reasoning]
- Therefore,
*dE/dt = 0*- The rate of change of the total energy of the system = 0

**Proof of Conservation of Energy in Quantum Mechanics**- For any operator
*Q, d/dt ⟨Q⟩ = -i/ℏ ⟨[Q,H]⟩,**where*represents the total energy of a system*H*- The time rate of change of the expectation of an operator
*Q***-i/ℏ**times the expectation of the commutator ofwith*Q**H*

- The time rate of change of the expectation of an operator
*[H,H] = 0*- Every operator commutes with itself

- Therefore,
*d/dt ⟨H⟩ = -i/ℏ ⟨[H,H]⟩ = 0*- The rate of change of the expectation of the Hamiltonian = 0

- For any operator