Conservation

# Syntax # Overview

Energy can not magically appear or disappear. It must go SOMEWHERE. We assume that energy obeys the Law of Conservation of Energy. The Law of Conservation of Energy states that energy cannot be created or destroyed, but can change its form. This law is in association with the total energy E of a system. The total energy of a system is the sum of all the energies with in a system, including its mechanical energy, thermal energy, or any external or internal energies. The total enery of a sytem changes due to the amounts of energy that are transferred to or from the system.

# Equations

## Equation One

Work:
W=sum of E= sum of E(mechanical) + sum of E(thermal)+ sum of E(internal)
Included in E(mechanical) are changes in kinetic energy and potential energy(elastic, gravitational, etc.)

## Equation Two

Isolated Systems:
0= sum of E(mechanical) + sum of E(thermal) + sum of E(internal)

## Equation Three

Power:
P(average)=sum of Energy/ sum of time
This occurs when an amount of energy E is transferred in an amount of time t.
P=dE/ dt
This is for instantaneous power.

A 2.0kg package of tamale slides along a floor with speed v1=4.0m/s. It then runs into and compresses a spring, until the package momentarily stops. Its path to the initially relaxed spring is frictionless, but as it compresses the spring, a kinetic frictional force from the floor, of magnitude 15N, acts on it. The spring constant is 10,000N/m. By what distance d is the spring compressed when the package stops?

E(mechanical1)=K1 + U1= .5mv^2 + 0 U=0
E(mechanical2)=K2 + U2= 0 + .5kd^2 U=0
.5kd^2= .5mv^2 - fd
5000d^2 + 15d - 16= 0
d= .055m= 5.5cm

A dog of mass m=6.0kg runs onto the left end of a curved ramp with the initial speed=7.8m/s at height y=8.5m above the floor. It then slides to the rigt and comes to a momentary stop when it reaches a height y=11.1m from the floor. The ramp is not frictionless. What is the increase E(thermal) in the thermal energy of the dog and ramp because of sliding?

E(mechanical) + E(thermal)= 0
sum of K= 0 -.5mv(initial)^2
sum of U= mgy - mgy(initial)
E(thermal)= .5mv(initial)^2 - mg(y-y(initial))= .5(6.0kg)(7.8m/s/s)^2 - (6.0kg)(9.8m/s/s)(11.1m-8.5m)
Approximately 30J

# Special Cases

1) Isolated Systems
There can be no energy transfers to or from a sytem that is isolated from its environment. In this case, the law of conservation of energy states that the total energy E of an isolated system cannot change. However, there can be internal energy transfers. This can mean kinetic, potential, or thermal transfers but the total energy E of all the forms of the energy may not change. In as isolated system, the total energy at one instant can be related to the total energy at another instant whithout vonsidering the energies at intermediate time.

2)Power
THe law of conservation of energy expands the definition of power. Power is the rate at which work is done by a force or power P is the rate at which energy is transferred by a force from one for to another.