Falling object experiment | Simple Application of Differentiation | Introduction to Newtonian Mechanics

     

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Introduction to classical mechanics

Inside... 

• Study on falling object experiment.
• How to calculate the acceleration due to gravity 'g' at any place.
• Equations of Motions.
• Introduction to Newtonian Mechanics

Falling object experiment     

     Take a stone or any object and drop it from the top of (say ten story) building or a tower.  You will notice that the object falls and hit the ground hardly in few seconds with high velocity, object may break into pieces due to large impact.  To get precise results from the experiment, it is advisable to drop slightly heavier objects to reduce the effect of air resistance.

Let us see what we can study from this simple experiment

     Let us do this experiment together and observe the event carefully.  Note that, even though we have just dropped the object initially with zero velocity (just by leaving the object), it hits the ground at some higher velocity (final velocity).  Thus the velocity has increased from zero to some higher value.  We call this increase in velocity as acceleration.  If you observe keenly, the velocity increases uniformly by increase in time.  In other words, acceleration is constant.  How can we measure the velocity of the object and say it is increasing uniformly?   How we say acceleration is constant?   What is the source of this Acceleration?  Let us see.

Study on Position

     Let us start talking about the change in position (displacement) of the object.  First, we observe and measure the positions of the object at each and every consecutive seconds during the fall.  By plotting the measured positions (metre) versus time (seconds) on a graph we get positive part of a parabola like curve (shown in figure below).  Practically it is quite easier to measure the position of the object rather than measuring velocity using metre scale or any other instrument.  Anyway, after reading this article you can measure yourself the velocity and the acceleration as well, using simple instruments and measurements.

Note : The below graph is the exact plot of the falling object experiment for first two seconds.

Finding the value of 'g'

     Equation of Parabola like curve has a general form of quadratic equation  y = ax² + bx + c  (second - degree equation) and thus as per the characteristics of the above graph (i.e.,) from the observation of free fall of the object we derive the Position equation s = ut + ½at² + s₀  . (compare with  y = ax² + bx + c .)  This is General form of Position - time equation.  Here 'u' is initial velocity and 's₀' is initial position of the object.  But in our situation we just dropped the object (u = 0) and assume initial position as zero, therefore we can write position equation as s =½at².    If we observe carefully (the graph for example), during 1st second the object travels nearly 4.9 metres, at the end of 2nd second it reaches more or less 19.6 metres from the drop point and at the end of 3rd second it reaches roughly 44.1 metres from the drop point.  Apply this measured displacement and time taken of the object to reach a particular position during our experiment in the equation s =½at² .  After the calculation, we get the acceleration value  a = 9.8 m/s² (metres per second squared).  Thus, we can determine the value of acceleration due to gravity 'g' in this way using Falling object experiment.

Study on velocity

     Let us discuss about velocity.  Recall that the velocity means displacement (change in position) divided by time taken during any interval of time.  From the above position - time graph and observations, we see that the displacement of the object during every consecutive second are not equal and increases quadratically.  Correspondingly, during every interval of time, the velocity increases uniformly.  To find the velocity of the object at an instant, we have to find the slope of that instant in the position - time graph.  But how can we determine the slope at an instant?  Even though there are various ways to calculate the velocity, differentiation is considered as an easy way to determine the velocity.  Differentiation is nothing but finding the rate of change (slope) of function.  Moreover, it gives us slopes of the position - time graph the whole event, from which we can find the velocity at any instant of the event.  As a result, we obtain a linear function  (y = mx + c)  graph and its equation is v = u + at . (see the graph below).  This is general form of velocity - time equation.  Here 'u' is initial velocity but in our case it is zero as we saw at the beginning and acceleration 'a' is acceleration due to gravity 'g' v = u + gt .  Therefore using the value of acceleration due to gravity, we can calculate the velocity.

Note: The above graph exactly shows the velocity at different instances of the falling object experiment for first two seconds.

     

Study on acceleration due to gravity

     Eventually, we came to important reason behind the falling object experiment.  It is none other than acceleration due to gravity.  It is the only reason behind the fall of the object.  However, the source of this acceleration is the gravitational force between the object and the Earth.  The gravitational force between them remains the same on or near the surface of the Earth or any planet during the fall.  From F = ma we can say that when force is constant the acceleration and mass remains constant.  Therefore, the acceleration of the object remains (almost) constant everywhere on or near the surface of the Earth of any planet.

To think : If you drop some objects of different masses, they all will hit the ground simultaneously.  Somewhat non-intuitive right?  Continue reading for the answer. (Important: Don't drop feathery, floaty or light weight object because air resistance will slow down their fall rate.)

     Resuming the functions and graphical discussion of our falling object experiment.  When the velocity increases uniformly the acceleration remains constant.  By differentiating the linear function  v = gt  we get  a = g.  This is a constant function (see below graph).  In other words, the slope of velocity - time graph is same all over the event, hence the acceleration has a constant value.

Note: The Above graph is exact plot of the acceleration value for first four seconds. 

     We can also use integral calculus to find the velocity from acceleration, and displacement from the velocity.  Because integration is inverse operation of differentiation.  Anyhow, we can use any of them to study motions.

     Hereafter, whenever you play with stones or any stuffs by throwing up and down, observe the above discussed matters in that and enjoy the Physics.

Equations of motion

     Now, we have three major equations to calculate displacement, velocity and acceleration of the falling object at any instant.

1. v = u + gt 
2. s = ut + ½at² 
3. v² = u² + 2as  (derived from first two equations)

     

Introduction to Newtonian Mechanics

     The above experiment is a part of revolutionary Galileo's Leaning Tower of Pisa experiment in the history of science which was done by Galileo Galilei between 1589-92.  Before him no one have concluded that Every object on earth falls freely at same rate (acceleration) irrespective of their masses.  Many of you might have heard this, if you drop a stone and a feather from same height, both will hit the ground simultaneously.  Because the magnitude of gravitational force will be different on different objects.  In other words, the gravitational force is adjusted to have constant acceleration.  Therefore, we say acceleration due to gravity is same for all objects on earth.  Actually due to air resistance in our atmosphere they won't fall simultaneously.  So we need vacuum to prove Galileo.  Watch the below video and enjoy.

     This revolutionary discovery paved a path to Great Mathematician Sir Isaac Newton to give another great revolutionary discovery, Theory of Gravitation.  To extend the theory of gravity to the Planets and stars Theory of indefinitesimal calculus was developed by Isaac Newton and Gottfried Wilhelm Leibniz in the later 17th century.

     And from the Newton's laws of motion, we realize that the force is said to be acting on an object only when the object accelerates.  If the object is moving in a constant (same) velocity or being stationary then the resultant force (sum of all forces) acting on it is said to be zero.  From this definition we can explain all three laws of motions.

     At the conclusion, if we understand Laws of Motions, the most fundamental part of physics then we  can understand most of the other parts of physics.  And also we can understand deeply how the things work.

Althaf Ahmed

India 

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