In the following few minutes, we're going to make a very brief introduction into classic
mechanics and forces and motion. Let's begin with a
historical overview. In ancient Greeks time
the relation between forces and motion was first
noticed and stated by Aristotle and his work "Physics".
and it can be summarised with the simple statement "motion requires force".
Which is practical, because if you have a cart, for example, and you want it to move with a certain velocity
V, you also need a horse, something that would apply a force
F to it. And as long as the force is applied, the cart is moving
as soon as the force or the horse stops,
the cart also stops. But of course, this view is wrong
and we actually know kinds of motion that is possible
without any external force. Probably the best example is the motion
in space where objects are isolated from external forces
within some approximation. And once accelerated to a certain velocity,
V, they continue
moving without any external force applied,
which is nice because you just need to accelerate, if you wants to get to
Mars, for example, and then decelerate, and you can save fuel. But it also means that,
for example, if you want to accelerate in space, you need something
to push off. And if you're ever going
space, don't forget to bring something with you, too, like a hammer, for example.
OK, so this new view on the relation between forces and motion
was combined and stated by Isaac Newton in his famous
Newton's laws of motion. Let's
state them in the mathematical form. The first law states that
the object is moving with a constant velocity if
either there are no external forces acting on it or
all external forces are balanced, which means that the net force
is exactly zero. The second law relates
the force acting on an object to its acceleration A.
Please note that this equation is vectorial,
which means that not only the numbers, not only the values of the physical
values of force and acceleration are related, but also their directions.
And the linear coefficient between them is known as the inertial mass.
Two important notes. The first one: both these
laws are only valid inertial frames of reference,
which should be defined by themselves. But it is sufficient to say that,
for example, the frame of reference bound to Earth is
a good approximation for an inertial frame of reference for basic
daily life processes on Earth.
The second one is less trivial and it is related to the range, the field of
applicability of classical mechanics. There are two limits
where it is no longer applicable. So the two conditions
must be satisfied for Newton's mechanics to be valid,
to be working correctly. The first one is related to Einstein's relativity theory.
The corresponding velocity of the object must be much lower than the
speed of light. And the second one is even less trivial. It says that
the characteristic size of the process in the phase space
should be much larger than Planck's constant.
So both these conditions will be actually discussed when you begin
studying field theory, relativity theory and quantum
mechanics respectively. OK. So what kinds of
forces are known? Let's make a short nonexclusive list, beginning
with gravity, the force attracting two masses M1 and M2,
which is proportional to their product and the inverse squared
distance. And it's also proportional to the universal gravity constant.
Then there is a similar Coulomb force either
attracting or repulsing two electric charges Q1 and Q2, which
is also proportional to their product and the inverse squared
distance between them. Then there is a corresponding force
related to the magnetic field known as the Lorentz Force, acting on
electric charges. Then there is reaction.
The force acting on two solid objects,
in mechanical contact, preventing them from passing through each other.
There is tension, elastic tension, proportional to elongation
of, say, an elastic spring or a rope according to Hooke's
law, then there is friction, which also comes
in different kinds. For example, this is the formula for dry friction
Proportional to the force of reaction. And
then there is viscous friction, which is, within some approximation, proportional
to the velocity of an object propagating in some viscous environment.
Then there is the force of pressure. Proportional to the pressure.
Which is basically the driving force of pneumatic devices.
OK. So almost all mentioned kinds of force
actually belong to the same basic type of interaction.
All of them, except for one, gravity, which is by itself a basic type
of interaction. In fact, all the others have electromagnetic origin,
electromagnetic nature. They fall into one category of basic types
of interaction. It can be reduced to Coulomb and
Lorentz forces. And actually, there are two
more basic types: strong and weak interaction
that you will learn about from quantum chromodynamics
and nuclear physics later. So since
we listed kinds of forces, let's say a few words about acceleration,
which is linked to the force. So it's a purely mathematical or
geometric description of motion. It's a kinematic
value. By definition, it is represented as the
first derivative of velocity with respect to time.
On the other hand velocity can
be expressed as the integral of acceleration over time with the reverse operation.
The same applies to the pair of velocity and the radius vector (the position of the
objects). Velocity is the first derivative of
the position, the radius vector, in time and radius vector can be found as the integral
of velocity over time. So if you know what are the forces acting
on the object at any point of time, you can reconstruct its trajectory.
Or you can reconstruct its motion. You can find its
position at any given point of time. For example, in the case where
the acceleration is constant, this integrals can be easily taken.
And then the radius vector as a function
of time will be given with this simple expression
in the form of vectorial equation. This equation can be projected
using the system of coordinates. It can be represented as a system of equations
on the coordinates. For example, this will be the equation on the coordinate X. You can write
down similar equations on the coordinates Y and Z. Let's
solve a simple example to show what kinds of problems you can solve knowing only this.
let's consider a cannon shooting a cannon ball with a certain
initial velocity V0. Let's
define the angle at which
it shoots the cannonball as Alpha, it's the angle to the horizontal line, to the
to the earth surface. Let us neglect its
curvature and say that it's flat to some approximation. And let's also introduce
a system of coordinates X and Y with the origin, the centre,
exactly at the point where the cannon stands. So the cannon ball will
follow some trajectory and will eventually fall down onto the
Earth's surface. Let's see if we can find out what would be the trajectory.
So we know that there is only one force acting on
the Cannonball, which is the force of gravity directed
at the centre of Earth.
In our coordinates this direction will be the direction down,
perpendicularly to the earth surface, and the force can
be represented simply as the product of the cannon ball mass and
the gravity constant g, which combines the mass
of Earth, its radius and the universal mass constant. For us,
it is important that g is a constant, which means that,
combining these with the second Newton's law,
we can notice that a is also a constant, which is exactly equal to g,
which means that the cannon ball motion is uniformly accelerated.
And this means that we can write down the equation of motion for
it in the form of a system of two equations on the coordinates X and Y.
Now, notice that G only enters the equation of
equation on the vertical
coordinate. Now we can exclude
time from this system of equations. For example, we can express t
from the first equation in terms of coordinate X and then substitute it
into the second equation on the coordinate Y. And we get
the relation between the two coordinates Y and X, Actually, geometrically,
This is an equation of a parabola. So it means that the trajectory
with some approximation neglecting friction, viscous friction in the air,
the trajectory is parabolic. Let's see if we can also find
the distance covered by the cannonball
to find it. We must find the second point
where the cannonball crosses the line Y
equals zero. So, the condition on the point where it falls onto the
earth surface is Y equals zero. So substituting Y equals zero into
the into this equation and solving this as an equation
on X, we immediately get the answer on X and the distance itself.
We also can apply a
trigonometric relation that two times sine alpha times cosine
alpha equals sine of two alpha. And
this way we get this short, simple answer.
From this expression, we can notice that
we can, for example, find the maximum possible distance.
Since sine cannot exceed one, the maximum
distance is exactly V0 squared divided by g. And it is reached
when Alpha is exactly forty five degrees. So if you want the maximum
distance, the cannon must be at exactly 45 degrees
to the horizontal line. So this
is a kind of problem you can easily solve, knowing only Newton's laws of motion,
simple kinematics and a bit of trigonometry.
Thank you for listening!
