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Scalars, Vectors, and Mysterious Forces

 

08-25-15
It is said that everything in the world (and the universe) can be measured by either a scalar or a vector.  A Scalar quantity is defined by an amount - or how much - of what is needed to be measured, and a Vector is defined by an amount and a direction.  Therefore a vector has both a scalar (or magnitude) and a direction. 

How much I decide to spend today, perhaps $35, is a scalar quantity.  I need only to state the amount to define how much I am spending.  If I drive an automobile to the bank, on the other hand, I will be travelling at a certain speed and in a certain direction.  I will need to state my speed in mph and the direction as either north, south, east, or west.

 

             

                      Scalars                                                 Vector 

A scalar quantity is defined as a numerical amount, or magnitude.  A vector has a magnitude and a direction.

Scalars quantities are quite obvious to us.  We know we only have to state a number to define how many matchsticks we have or how many cups of flour for a recipe.  Vectors, on the other hand, are perhaps not so obvious in that a direction is always stated along with the amount, or magnitude.  Vectors are used to define quantities whenever a direction is present such as velocity, acceleration, or force.  Without direction a vector quantity is never considered complete.  This may seem somewhat a matter of trivia that a direction is always needed to complete the description of a certain quantity – isn’t the amount the main thing ?  How much of something may be vital to some kinds of data, but when it comes to quantities such as velocity or force, the direction is just as important as the amount. 

This is especially true when it comes to motion around a curve.  An object can be travelling at a constant speed around a curved path but its direction always changing:

 

The vector quantities called Velocity in the above sketch have the same speed but change direction.

The speed of the object is the same – which is a constant scalar amount – but its direction is changing and introduces a different predicament where a new force is present called Centrifugal Force.  It is a force that is directed outward from the center of the arc of motion.  It’s as if this force has come out of nowhere only because we have changed our direction of motion.  But this force is real, and can be measured like any force in the physical world (or calculated).  Is Centrifugal Force considered a mysterious force that we must give special attention to and take time to understand ?  For some, centrifugal force is commonly experienced in our everyday lives.  An example is when one banks a left turn in their automobile and it tilts onto its right wheels.  But perhaps they did not realize this would occur since they did not change their speed, only their direction.  Natural philosophers such as Sir Isaac Newton pondered these kind of phenomena because it explained how planets remained in their orbits.  Centrifugal force is the force that counteracts gravity such that the planets will never fall into the Sun. 

Speed is constant around a circular path and causes a Centrifugal Force.

 

It’s as if this force has come out of nowhere only because we have changed our direction of motion.  But this force is real, ..

It is thought fortunate that some thinkers took a fascination for these type of phenomena even though others thought it only a topic of trivia.  In other words, it does not require a leap into the realm of quantum physics, for example, to venture the mysterious workings of nature.  For Newton, there was enough mystery in the phenomena of gravity.

The “mysterious force” called Centrifugal Force has shown to still hamper modern day train travel, where trains entering curves beyond a certain permissible speed all of sudden derail.  This occurrence doesn’t always seem to match our intuition, but it does certainly match physical law. 

Trains experience centrifugal force when going around a curve.

Another example is the aerobatic aircraft performing a loop maneuver which introduces a centrifugal force throughout the entire loop.  Depending on the orientation of the aircraft within the loop, the centrifugal force can either subtract or add to the weight force of the aircraft.  Near the very top of the loop the centrifugal force will be subtracting from the weight, but near the bottom when exiting the loop centrifugal force will be adding to the weight - and which will require enough lift from the wings to counter both the aircraft’s weight and the centrifugal force.  All aerobatic maneuvers should be carefully calculated because of the effects of centrifugal force. 1

  

The weight of the aircraft W and centrifugal force Fc subtract from one another at the top of the loop.  Upon exiting the loop (bottom), the two forces will combine causing the greatest downward force experienced by the aircraft during the loop maneuver.  (Vectors not to scale.  Lift force under wing not shown). 

        

1. Physicists also define a 'centripetal force' which acts opposite the centrifugal force with equal magnitude and satisfies a force balance required of Newton's Laws.