Theme 4 – Newton and Gravity

ASTR 101Prof. Dave Hanes

Newton – The Laws of Mechanics

[Forces and Motions]

His Objective

To develop a quantitative, mathematical and predictable understanding of how objects move when they are influenced by forces.

Newton’s First Law(Inertia)

A body in any state of motion (including

a body at rest) will stay in that state ofmotion unless an unbalanced force is acting on it.

Unlike Galileo, hegot this completely right!

“Acceleration” defined

In physics, Acceleration is any change in the state of motion of an objectThis includes speeding up, slowing down, and/or changing direction.

Newton’s Second Law (in words!)

If an unbalanced force acts on a body, it will accelerate (its state of motion will change)

The degree to which it does so depends on the size of the force (obviously!) but also on the mass of the object (how much material it contains, its sluggishness or resistance to motion).

The Colloquial Meaning of Mass

Massive headache Massive amount of work

But in Physics

Mass is a measure of an object’s resistance to being pushed around, and depends on how many atoms it contains in total.

May be unrelated to size!

The Law as an Equation (not essential here!):

F = m a, or equivalently a = F / m.

In other words (looking at the right-hand version):

For a given object of mass “m,” a bigger force (“F”) produces a greater acceleration (“a” = change in the state of motion)

but for a given force (“F”), a more massive object (larger “m”) is accelerated less (smaller “a”) than a lower-mass object

Try it! -- throw a baseball, then a shot put!

Newton’s Third Law

Every force (‘action’) is matched by a force

of the same size but in the oppositedirection (‘reaction’).

Often used metaphorically

Push the Wall; It Pushes Back

http://www.astro.queensu.ca/~hanes/ASTR101-Fall2015/ANIMS/THIRD-LAW.mp4

Car Breakdown

Why must you get out? If you don’t: You push the

dashboard It pushes back on you You are pushed into

the seatback That pushes the car

itself towards the rear So nothing moves! No

‘unbalanced’ force!

Note That Everything Moves[although perhaps imperceptibly!]

You push the car – and so the car pushes you, and through your feet, the Earth itself. (What would happen if you were on glare ice?)

As the car moves forward, the Earth moves back – but only a microscopic amount, as it’s so massive!

Likewise pushups – you go up, the Earth goes down!

GravityA New, Very Special Force

As Envisaged by Newton

‘action at a distance’ – no contact required, no strings, no pushing or pulling

universal (acts between any two lumps of matter in the Universe)

inverse-square (e.g. weaker with distance, in a very precise way. Two objects 2 metres apart feel just ¼ of the force they would feel if they were 1 metre apart)

Why Inverse-Square? Analogy to the Intensity of

Light

…or to a paint sprayer!

Like an Apple, the Moon is Continuously

Falling!

Its sideways motion keeps it

from landing on our heads!

For every 1 km of sidewaysmotion, it falls about 1.3 mmtowards the Earth. So it

movesat a nearly constant distanceaway, in a near-circular orbit.

Calculating the Force of

AttractionHow do you add up the tug every atom in one body exerts on every atom in the other body?

Very challenging, especially for irregularly-shaped objects.

Newton’s solution: invent calculus.

An Amazing Simplification

Consider a “spherically symmetric body” -- one that

looks the same no matter what direction you drill into it

Not this But this

And this

…and These Astronomical Objects!

As If by a Miracle!

So All You Need to Know:

What is the total mass closer to the centre than you are? (i.e. add up all those atoms)

How far are you from the precise centre?

Bingo: out pops the force in question.

Why This Simplicity?(Nature’s Gift to Us!)

It’s because

1. Gravity obeys an inverse-square law; and

2. Big objects are spherical

Not Vacuum Cleaners!

If the sun became a black hole (which it won’t, by the way!):

Its gravitational effect on the Earth would be unchanged: we would not be ‘sucked into’ it.

We would continue to orbit it, once a year, just as before – but in perpetual darkness!

Planning Space Flight Made Easy!

We coast past planets, in rockets and space probes

Calculating the forces, and the motions, is sublimely easy!

How Do Things Move Under Gravity’s Influence?

A ‘gedanken’ [thought] experiment, not easily implemented in practice.

One Problem:The cannonball will hit the ground!

Solution: Shrink the Earth right afterwards!

Low-Speed Cannonball

The cannonball fallstowards the (shrunken)Earth, picking up speed. It whips around it,

movingreally fast, then climbsback to the starting

point,losing speed.

This orbit repeats! It’s asmall ellipse, with theshrunken Earth at the farfocus, the former centre.

High-Speed Cannonball!

The cannonball moves away

from the blue Earth alongthe curved path shown,losing speed all the while.After reaching a maximumdistance (at the bottom ofthe figure) it starts to fall back, picking up

speed.

The orbit repeats. This time,

it’s a huge ellipse, with the earth at the near (top)

focus.

At the “Just Right” Speed

The cannonball movesaround the Earth in a

perfectcircle (a special kind ofellipse!) at constant

speed.

This is what the ISS(International Space

Station)does, by our design.

A Full Understanding

The planets are orbiting the Sun under theinfluence of its gravitational attraction.

They are obeying Newton’s laws of motion,accelerating and moving because of thegravitational forces they feel.

Newton was able to explain all three of Kepler’s

Laws!

Beyond Kepler’s First Law

Newton asked: What would happen if we shoot the cannonball really really fast?

There are two answers to this.

“Escape Velocity”

If we launch it at just the rightspeed, the cannonball coasts

offalong a curve known as a

parabola,never to return.

Eventually it will be barely inching

along, having lost essentially all of

its speed. It has only just enough

energy to escape, will never quite

stop, and will never return to the

Earth.

Faster Still?

If we give it more thanbare escape speed, thecannonball moves offalong a curve knownas a hyperbola.

In this case, it will still be moving with considerablespeed even when far away.

Putting These All Together

Planning the Path!

The final path or orbit depends on the speed with which you launch (that is, the energy) and the initial location and direction.

Are You Geometrically Perplexed?

Kepler discovered elliptical orbits (including circles) for bound objects (those in repeating paths).

Now Newton has introduced parabolas and hyperbolas for unbound objects (those that will escape to infinite distance).

No problem! they are mathematically related in a very direct way, as Newton showed. The details needn’t concern us. (See the next panel if keen.)

Conic Sections

Make a cross-sectional cut through a cone.

The curve that results depends on the angle at which you make the cut.

These curves are closely related mathematically.