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Theme 4 – Newton and Gravity
ASTR 101
Prof. 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 of
motion unless an unbalanced force
is acting on it.
Unlike Galileo, he
got this completely
right!
“Acceleration” defined
In physics, Acceleration is
any change in the state of motion of an object
This 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 opposite
direction (‘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!
Gravity
A 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 sideways
motion, it falls about 1.3 mm
towards the Earth. So it moves
at a nearly constant distance
away, in a near-circular orbit.
Calculating the
Force of Attraction
How 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
1.
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 falls
towards the (shrunken)
Earth, picking up speed.
It whips around it, moving
really fast, then climbs
back to the starting point,
losing speed.
This orbit repeats! It’s a
small ellipse, with the
shrunken Earth at the far
focus, the former centre.
High-Speed Cannonball!
The cannonball moves away
from the blue Earth along
the curved path shown,
losing speed all the while.
After reaching a maximum
distance (at the bottom of
the 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 moves
around the Earth in a perfect
circle (a special kind of
ellipse!) 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 the
influence of its gravitational attraction.
They are obeying Newton’s laws of motion,
accelerating and moving because of the
gravitational 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 right
speed, the cannonball coasts off
along 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 than
bare escape speed, the
cannonball moves off
along a curve known
as a hyperbola.
In this case, it will still be
moving with considerable
speed 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.