Energy Diagrams

Conservation of Energy

Elastic Potential Energy

Main principles

Potential Energy

Gravitational Potential energy

Forces

You're driving at 35 km/h when the road suddenly descends 15 m into a valley. You take your foot of the accelerator and coast down the hill. Just as you reach the bottom you see the policeman hiding behind the speed limit sign that reads '70 km/h." Are you going to get a speeding ticket?

With what minimum speed must you toss a 100 g ball straight up to just touch 10-m-high roof of these gymnasium if you release the ball 1.5 m above the ground? (solve using energy)

What is the speed when it hits the ground? (if you didn't catch it)

12.9 m/s

14 m/s

A pendulum is made by tying a 500 g ball to a 75 cm string. The pendulum is pulled 30 degrees to ones side then released.

What is the ball's speed at the lowest point of its trajectory?

To what angle does the pendulum swing on the other side?

1.4 m/s

30

A. d>a>c>b B. c=d>a=b C. d>c>a>b D. d=c=b=a

A 1500 kg car traveling at 10 m/s suddenly runs out of gas while approaching the valley shown here. The alert driver immediately puts the car in neutral so that it will roll. What will be the car's speed as it coasts into the gas station on the other side of the valley?

1.41 m/s

**Energy diagrams**

As noted earlier, one of the tactics to solving

these problems is finding the initial energy, and have that be conserved.

One way of visualizing this is an energy diagram

Most of this book's little diagrams, I don't agree with,

This one however works well.

This first one is of a gravitational potential only

This is a spring potential

This one is even more general, it could be any type of potential

This brings up the idea of Equilibrium positions

An equilibrium is where an object doesn't move.

or where the forces are balanced.

There are multiple types

Stable, and unstable as well as others.

**how many equilibrium**

are there in this graph

A. 1

B. 2

C. 3

D. 4

are there in this graph

A. 1

B. 2

C. 3

D. 4

**How many of these**

equilibrium are

stable?

A. 1

B. 2

C. 3

equilibrium are

stable?

A. 1

B. 2

C. 3

To be stable, a small perturbation from the equilibrium will

cause the object to return to that equilibrium

Unstable is when a small perturbation leads to greater movement

away form the equilibrium

Now we need the potential energy of a spring.

Note it is an energy of position, not motion, thus it is a potential

energy, specifically elastic potential energy.

Again make note of the delta x. This is a change in position from the

equilibrium. It can thus be either a stretching or a compressing of the

spring.

Because the term is squared, the sign turns positive no matter what.

Note it is also a scalar, no direction, and the units are Joules

Finally I ask, can you have a negative potential energy of a spring?

When a spring is horizontal the only energy

is kinetic and potential of the spring.

where as when it is vertical the potential energy

can be store both in the spring and due to gravity.

there is an example in the workbook (8)

lets solve this problem. With spring constant

k = 250 N/m

m = 5.0 kg

h = 40 cm

Also our little parable, only had one type of energy, Mechanical

He transferred his savings (potential) to liquid (kinetic) energy.

Mechanical isn't the only type of energy.

There is an internal energy of the motion of each atom, this is a

thermal energy, it has to do with temperature.

Thus we define energy of a system now as:

With this addition of thermal energy we can transfer our total

energy around a bit more.

We can explain why a book comes to rest.

The initial kinetic energy, got turned into thermal energy.

It heated up the atoms.

As we consider our basic model of energy we now can discuss decreases to our system, Not just straight conservation inside the system.

Looking at this. We can say energy is transferred in

to our system if W is positive. (out if negative)

and

Energy is transformed in our system if W is zero.

Think back to the text book on table.

Tell me what happens to each of these terms and

what does that mean for my system.

Can you think of a similar example with Work being

the opposite sign?

We worked through this many time last chapter.

If we add up all forces and directions we get the following change in kinetic energy. (make sure it is net work)

Not a lot of change, but important none the less. This is the

Work-Kinetic Energy theorem.

Consider a block sliding down a hill

The force here is due to gravity

Substitute, this cosine theta ds = dy

and get

or a final version with out small steps:

Note this work is independent of the path that

it takes to get there.

I could have dropped the block of a cliff of same

height, and gravity would do the same about of

work.

So a force that is path independent is called

a conservative force.

Now this was just gravity, but any conservative

force that is path independent works

you'll also notice that the Work is equal to a

change in potential energy.

This potential energy came from a conservative

force, Again any conservative force.

Look at a spring.

This brings up the following result

More specifically:

Where Work is made up of both conservative

and Nonconservative parts

From this result and the work kinetic

energy theorem we get

From this we see Mechanical energy is conserved

if there are only conservative forces

On board, graph both potential energy to gravity and

spring potential.

Before we leave this analysis let us look at one more thing

This says that the derivative of potential energy with respect

to position is the negative force.

or think of a slope of energy vs position.

We previously broke Work into conservative and

non conservative parts

Let us take that one step further and break the

non conservative work into

Dissipative

External

External forces are just that.

For example, pushing a book, or pulling a sled

I ask: are they really non conservative?

Answer: yes, just think about the path I would lift the book

Dissipative forces give rise to dissipative work,

these come from any form of drag: friction, air

resistance, etc.

The only trick here is the system.

We need all of it, for this term, both the book and

the table, both experience friction, and hence

dissipation.

So that makes are work turn into these three

making some substitutions

Work- kinetic energy theorem

Work & conservative forces

Thermal energy and dissipative forces

Here we have the total energy of a system

is equal to the external work being done to it.

If the system is isolated, then the total energy

is conserved. Granted the energy can be transferred

between different parts.

In perhaps a little more user friendly form

From here it is all about before and after shots.

Find what you want to call initial and final, and

go from there.

Potential energy due to gravity

Could you explain the gravitational potential energies of the falling balls in the stop to think example?

Can you please explain potential gravitational energy?

How do we choose the correct system to work with?

A 7.0-kg rock is subject to a variable force given by the equation

F(x)=6.0N−(2.0N/m)x+(6.0N/m2)x2

If the rock initially is at rest at the origin, find its speed when it has moved 9.0 m.