Friday, May 11, 2018

Unizor - Physics4Teens - Mechanics - Dynamics - Superposition of Forces

Notes to a video lecture on

Sum of Forces

Previously, in most cases we were analyzing the motion of objects with only one force participating in the process.

With few exceptions our objects moved either along a straight line with a
force acting along this line or rotated around some center with only
one force keeping them on their circular trajectory.
A couple of problems related to launching of projectile at an angle to
horizon did deal with a more complex motion, but even in this case there
was only one force - gravity - that acted on an object.

This lecture is dedicated to general approach to dealing with multiple
forces of different types acting simultaneously on the same object.

Let's review what kind of forces occur in real life and, therefore, might affect the motion of an object.

Applied Forces

These are forces caused by one object contacting another (pull, push, roll etc.).

These are the first kind of forces people dealt with and understood
relatively well in a sense that they always saw the source of them.

A car moves because its engine pushes the wheels forward. A person walks
because his muscles make his legs move in a special way to push back
the road.


The effect of friction is slowing down the movement, unless additional
forces are applied. The source of friction is non-ideal contact between a
moving object and its surrounding, that causes tiny bumps between them
catch each other, thus preventing the uniform motion.

To the same category we can put resistance of air, when a plane flies in
a sky or resistance of water, when a submarine moves in the deep sea.


If you pull a cart by a rope, you pull the rope, not a cart. That force
somehow is transferred to a cart. The mechanism of transferring a force
from one end of a rope to another is tension. Your force of
pulling attempts to stretch a rope, but internal links within a rope do
not allow it to stretch, one link is attached to another and transfers
the force down to all tiny links inside the rope. That causes the rope
to become tense, and the force you applied to one end of the rope is the
cause of the rope's tension, which, in turn, pulls the cart.


Elasticity is a property of an object, like spring or rubber, to change
its form under some external force and then restore the original form,
when the force is no longer applied. If an object stretches with its
weight a vertically positioned spring with fixed top end, reaching some
fixed balanced position, it is the elasticity force of a spring that
counteracts the force of the object's weight, thus holding an object in a
fixed position.

The cause of elasticity is in the inner structure of a material. Steel
of spring is such an elastic material, as well as rubber and others.


Any two masses attract ("gravitate") each other. The attraction is
mutual, the force of gravitation of object A towards object B is
directed from A to B and is equal in magnitude, but opposite in
direction, to the force of gravitation of object B towards object A,
directed from B to A.

Earth attracts everything on it, like people, houses, airplanes in the
sky, ships in the sea, while each of them attract Earth with the same in
magnitude, but opposite in direction force.

Quantitatively, the force of gravitation between two masses proportional
to each mass and inversely proportional to square of distance between
them. This is an experimental fact, we'll discuss it later in this


This force is in some way similar to gravitation, but in case of
gravitation we have only one type of force - attraction, while in case
of electricity it might be attraction or repelling, depending on what
kind of electrical charges we deal with.

Two electrically charged objects with opposite charges (which we call
"positive" and "negative") attract each other similarly to gravitation.

Quantitatively, the force of electrical attraction between two
electrically charged objects is proportional to each charge and
inversely proportional to square of distance between them. This is an
experimental fact, we'll discuss it later in this course.

If two positively or two negatively charged objects are involved, the
forces will act in opposite direction, the objects will be repelled from
each other with the force of repelling exactly the same in magnitude as
the force of attraction in case of two opposite charges.


This force is in some way similar to electricity, but in case of
electricity we have two types of objects - positively and negatively
charged, while in case of magnetism each magnetized object has two ends
- two "poles" with opposite "polarity". There are two kinds of
polarity, that we conditionally call "North" (N) and "South" (S), and
opposite ones (N - S) attract each other, while similar ones (N - N or S
- S) repel.

Two magnetized objects attract or repel each other, depending on which side of one object is close to which side of another.

Quantitatively, the force of magnetic attraction or repelling between
two magnetized objects is proportional to degree of their magnetizing
and inversely proportional to square of distance between them. This is
an experimental fact, we'll discuss it later in this course.

So, there is a multitude of forces that can act on our object
simultaneously and we have to be able to analyze their combined effect
on the motion.

The main idea of a motion, when multiple forces act on an object, is a principle of superposition of forces.

It is experimentally confirmed and accepted in classical Physics as an
axiom that, if two or more forces act on the same object at the same
time, the object moves identically to the movement caused by only one
force, that is a vector sum of all individual forces acting on an

This axiom is called Principle of Superposition of Forces.

Here we are talking about general principle, that we accept for any kind
of forces in three-dimensional world, each being a vector: acting together, these forces produce the same effect, as if they are all replaced by one force - their vector sum.

Let's illustrate this with a couple of examples.

1. When a rocket is launched vertically up, there are two major forces
acting upon it: the force of gravity pulls it down and its engine pushes
it up (stronger than gravity, of course). These two forces are directed
along the same vertical line, but opposite in direction. Their vector
sum is directed upwards and quantitatively is determined by a vector sum
of these major components.

2. A car moves up the hill and is acted upon by three forces:

gravity pulls it downward,

its engine pulls uphill and

the reaction of the road pushes perpendicularly to the road.

The vector sum of these three forces must point up the hill.

3. An object is hanging on a thread fixed to a point on a ceiling and is moving in a vertical plane like a pendulum.

There are two forces acting upon it at each moment:

gravity pulls it downward,

tension of a thread pulls it towards the point on a ceiling, where a thread is fixed.

The vector sum of these two forces must be tangential to an arc of
trajectory of this object and directed towards this arc's midpoint.

All the above examples are examples of the Principle of Superposition of Forces
- a universal principle of Physics, which we accept as an axiom,
regardless of the physical nature of the forces involved, and which is
confirmed by experiments.  

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