## Work

...a connection between force and energy.

Some vague relationships:

- Energy $\leftrightarrow$
'damage' $\leftrightarrow$ change in environment

Does a force **always** cause a change in the environment??

### Work

Forces?

When do forces cause a *change in the environment?*

- When a
**force**is exerted an object, - and the object
*moves*.

Work = force $\times$ distance $= F \times d$

Shelf -> book | Bowstring released -> arrow | |

Force on an object? | ||

Work happened? |

### Work and Energy

Work is a transfer **of energy**.

Units?

- Measuring force in Newtons,
- distance in meters

$\Rightarrow$: 1 Newton $\cdot$ meter $\equiv$
1 **Joule** $=2.78\times 10^{-4}$ Watt $\cdot$ hours $=2.39 \times 10^{-4}$ **C**alories.

### Example: How much work?

How much work must you do to lift a 10N book (with a constant speed) up from the floor, to
a point 2 m above the floor?

W = 10 N $\times$ 2m = 20 N$\cdot$m = 20 Joules.

### Gravitational energy

Does a raised book have energy? Have the ability to cause destruction in its environment? How much destruction?

We think this **gravitational energy** depends on... mass $m$,
height $h$, gravity's acceleration $g$.

**Guess** a formula for gravitational energy...

$\frac{mg}{h}$ | $\frac{g}{mh}$ | $\frac{1}{mgh}$ |

$mgh$ | $\frac{h}{mg}$ | $\frac{mh}{g}$ |

### Gravitational energy

The answer is

$${\text GravE} = mgh$$

Notice that this is precisely how much work you need to do to lift the book when you pick it up directly:

$${\text Work}=F\cdot d =(mg)\cdot h$$

In the pulley lab, different pulley arrangements allowed you to lift with a different force. But the smaller the force $F$, the longer the distance you had to pull the string $d$.

If the lab went well, many of you found that

$$F\cdot d \approx mgh$$

The work you did on the weight was equal to the gravitational energy gained by the weight.

So the weight-on-a-string is like a **gravitational battery**:

- Lifting a book $\Rightarrow$ energy in.
- Dropping a book $\Rightarrow$ energy out.

### Ability to do work

We say a system has **energy** if it has the **ability
to do work**.

### Energy of moving objects

It seems that moving objects also have the ability to do work.

How does energy depend on...

**mass**$m$ of the moving object?**speed**$s$ of moving object?

$ms$ | $\frac{m}{s}$ | $ms^2$ |

$\frac{s}{m}$ | $\frac{1}{sm}$ | $\frac{3}{sm}$ |

### Kinetic energy

The energy of a moving object is
called **kinetic energy** and is equal to:

KineticE = $ \frac{1}{2}ms^2$

Many
more lives are lost each year due to **kinetic** energy than due to **nuclear** energy!

### Conversion of energy from one form to another

An
amazing observation... with no friction or air resistance, the
**gravitational energy** of a book just before dropping turns out to be
precisely equal to the
**kinetic energy** of the book the instant before it hits the floor.

$$mgh =\frac{1}{2}ms^2$$

It is as if the gravitational energy was completely **converted **to kinetic
energy.

This gives us a way of calculating how fast a dropped object is moving:

$$\begineq mgh &=& \frac{1}{2}ms^2\\
gh &=& \frac{1}{2}s^2\\
2gh&=& s^2\\

\sqrt{2gh}&=&s\endeq$$

What is the speed of an object dropped from a height of 5 m (about 15 ft: the second story of the Ad building) just before it hits the sidewalk?

[Using $g \approx 10$ m/s^2, the number you get will have units of m/sec.]

How many miles per hour is that? Use WolframAlpha to find out...

### Energy CONSERVATION

The more general way to think about this is to say

Energy is never lost--only converted from one form to another

We write it and understand it this way... that you have to add up all the
possible **forms of energy**, and their **sum** is what never changes...

$$E_\text{ start} = E_\text{end}$$

$$GravE_{\text start}+KE_{\text start} = GravE_{\text end}+KE_{\text end}$$

$$mgh + 0 = 0 + \frac{1}{2}ms^2$$

$$mgh = \frac{1}{2}ms^2$$

Bring hotwheels car

### Suggested exercises

Conceptual exercises, Chapter 6: 5, 6, 12, 15, 16

### Image credits

NYCArthur, Al, Jason Rogers, Daniel Williams, Nicholas T, Tim Pearce, Chandru Ramkumar, Chris, Bu, Ville Miettinen, 4ELEVEN images, Andrea Williams