Newton's Laws

Galileo Newton

What we are trying to do:

First Law (Galileo's law of Inertia):

Up to now we have just described motion (kinematics). We now want to explain it (dynamics)
A Body continues at rest or in a state of uniform motion unless acted on by a force.
Uniform motion means no acceleration. Note forces can balance: "a force" means "a net force"

Inertial/Non-inertial frames

Inertial/Non-inertial frames: A body continues at rest.....

At rest with respect to what? Watch the animation!!

Inertial frames: moves at a uniform velocity w.r.t "fixed stars"
Non-inertial frames: body accelerates w.r.t. frame without forces being applied
A very common non-inertial frame is a rotating one: e.g. a record player turntable.

There are two frames of reference an inertial one (well almost) attached to the earth (x,y) a non-inertial one (x',y') The earth rotates, but is inertial to about 1/2 %. The only truly inertial frame is one defined by the distant stars.

Second Law of Motion

First we need to have concepts of force and mass. Note first that mass is not weight (which is a force!)
2 different objects have same force, different accelerations. Define \color{red}{\frac{{m_2 }}{{m_1 }} = \frac{{a_1 }}{{a_2 }}}

and define arbitrary object to have mass 1 kg.

Second Law

Then 2nd law reads

The single most important equation in Physics!

Force = mass x acceleration

Unit : [F] = [m] [a] = (M) (L T-2)

= kg m s-2

But so important that we need a new unit.

1 Newton = 1 kg m s-2

Note this is a vector equation:
\color{red}{ \vec F = m\vec a}

Third Law

Action and reaction are equal and opposite

An action is a force exerted by one object on another

The reaction is the force exerted by the second on the first

Note that it is particularly easy to forget reaction forces: in this case, if you ignore the reaction force, the block would fall through the table.

Examples of Forces:

Most important is gravitational force, which gives rise to weight 
m a = W = -mg

so that 

a = -g

Hence Newton's 2nd. law shows that everything has the same acceleration in a gravitational field. This is unique to gravitation, and suggests that there are two kinds of mass: inertial mass and gravitational mass.

Gravitational force depends on position: 

 F = mg

is a good approx. only at surface of earth. On moon, weight ~ 1/6 that on earth

Other Forces

Momentum and Collisions

There is a second (and better) way to state the second law

First define momentum= mass x velocity

\color{red}{ \vec p = m\vec v}

Impulse

Second law can be written

Force x time for which it acts = change in momentum

This quantity \color{red}{\vec F\delta t} is known as impulse, and is useful in collisions. However momentum also has a very important role in connection with the 3rd Law (Action and Reaction.............)
e.g a bouncing ball. Note that the force only occurs for a very short period of time.

If a 300 gm ball is dropped from 2 m, what is the force during the bounce if it takes 1/50 s?

Circular Motion I

Circular Motion: Body moving in circle at constant speed will accelerate towards the centre

Note constant speed does not mean constant velocity. e.g. consider a car travelling round a quarter circle, radius R, speed v

Car on a curve

    face face face
  • To find the change in velocity,
    face face face

Dimensionally: must be combination of v and r that gives dimension of accn.

  • [r] = L
  • [v] = LT-1
  • [a] = LT-2

so that only possibility is 

a = v2 = (LT-1)2
    r     L
= L2T-2 = LT-2     
   L

Centripetal Acceleration

This is called centripetal accn. It is NOT centrifugal accn. (which is the apparent accn. that a body feels in a rotating frame of reference)

Centripetal force is force required for centrip. accn. It does not exist as a force in its own right: it has to be supplied by another force: e.g.

Simplest example
A kid whirls a stone around which is tied to a piece of string of length 50 cm. The string has a breaking strain of 20 N, and the stone weighs 400 gm. How fast is the stone going when it the string breaks? Ignore the fact that the rope gets pulled down by the mass.

Newton's Discovery of the law of Universal Gravitation

We have the ingredients to understand the solar system
Need extra ingredient of Kepler's laws:

Law of universal gravitation

: applies between any two bodies anywhere in the universe
Gravitational force between any two bodies, masses M1 and M2 separated by distance R is given by Newton's Law of universal gravitation
\color{red}{ F = \frac{{Gm_1 m_2 }}{{R^2 }}}
Note that this imples that grav. force gets weaker as we move away from the earth
G=6.67x10-11 N m2 kg-2 is a universal constant
The first direct test of this was by measuring the deflection of a plumb-bob near a mountain in Scotland (Schiehallion)

Note that action and reaction apply: normally the reaction force will not cause a large acceleration.

The inverse square law is fundamental: there is an interesting and wrong way to get it.
The real derivation of Kepler's laws is much more complicated: if you are interested look at Celestial Mechanics

Energy

Newton's second law gives us a relation between velocity and force, via 
F = m a = m δv 
            δt
If the force is complicated (e.g. F = -kv2, or F = sin(r) ), then solving for the velocity can be very difficult. Fortunately there is a better way: a new idea called Energy.

Kinetic Energy

defined to be 

  T = 1/2 m v2
for a particle with mass m, vel v.

Potential Energy

If you drop something, kinetic energy increases. This energy is originally in the form of potential energy (P.E.).

Conservation of Energy

A very important idea that we will return to again and again:

Perpetual Motion Machines

An interesting consequence....
We would be able to arrange for a complicated system of forces to keep things moving forever.
Or another variety
and here is a modern version of the

Escape Velocity

For gravity in general, (not close to earth's), we get
\color{red}{ U\left( r \right) = \frac{{GMm}}{r}}
This doesn't look much like the usual form 
U(h) = mgh
but................

In fact the slopes are the same near the surface of the earth
face face

Escape Velocity

How hard would you need to throw something so that it never came back?

Angular Momentum

Keplers 2nd law

(the equal area law) is in fact just conservation of ang. mom.
Since 
L = mvr or vr = L/m
is a constant, so area of triangle 
δA = r vδt = L/m δt

is also a constant

Tops

The most baffling example of ang momentum conservation is a spinning top/gyroscope

Conservation Laws

A lot of material: what do we really need?