CALCULUS IN PHYSICS
MUGAMAD YAASEEN MALLIE
“”Mathematics is the language of physics”. When it comes to physics the use of mathematics in this field of study varies from trigonometry, algebra to geometry. Calculus plays a vital role in the study of physics as many laws and theories are derived and explained through the processes of calculus. In this assignment the use of calculus, with regards to integrational and differential equations in physics, will be illustrated with special references to Newtonian mechanics, electromagnetism, and thermodynamics.
Uses of integration, differentiations in physics
“Integrals and derivatives are literally used everywhere in the physics world. When calculus was discovered, its first real life application was first used in physics.”
“Newtonian mechanics describes how a force acts on a certain object as well as the how the object will move because of the force acting upon it (Benjamin Crowell, Newton’s “Axioms or Laws of Motion” page 19). It is believed that calculus was discovered by Isaac Newton himself and the first time calculus was used in physics was during the study of Newtonian mechanics.”
Example: motion-displacement, velocity and acceleration.
“The displacement of any object is described as the rate of change of the objects distance.
R(t)=?D?t , where R(t) = displacement at any time, ?t=change in time and ?D=change in distance
Velocity refers to the rate of change of the displacement of an object. This equation can be described as the first derivative of the equation of displacement.
The acceleration of an object is described as the rate of change of velocity per unit of time, also this equation can be described as the second derivative of the equation for displacement.”Rt”=d^2Ddt^2
“Electromagnetism is the study of electromagnetic forces that exists between two or more electrically charged sustenance’s (Fundamentals of applied electromagnetics (6th ed.), page 13 by Ravaioli et al).”
Example: Faradays Law
“This basic law of electromagnetism describes gives a prediction of how a magnetic field would interact in an electric circuit. Faraday’s law of induction makes use of the magnetic flux. The magnetic flux is defined by a surface integral”:
?B=?(t)Brt.dA” Where dA is an element of surface area of the moving surface ?(t), B is the magnetic field (also called “magnetic flux density”), and B·dA is a vector dot product (the infinitesimal amount of magnetic flux through the infinitesimal area element dA). In more visual terms, the magnetic flux through the wire loop is proportional to the number of magnetic flux lines that pass through the loop.”
Thermodynamics is the study of heat and temperature of an object as well as their relationship to energy and work (‘Theory of Heat’, Clausius Rudolf ).
“In terms of physics, “work” refers to the energy being transferred from one body to the next over a period of time. When it comes to real life applications, work is known to be happening under variable force. For a instant snapshot, work can be described as:
?W=Fx.?x where W= work, F(x)= force as a function of distance and x=distance.
If we had to add up work done by an object of an infinite period:
From this, we can define work as a definite integral of the force across any distance travelled:
For explanations of Newton’s laws of motion by Newton in the early 18th century, by the physicist William Thomson (Lord Kelvin) in the mid-19th century, and by a modern text of the early 21st century, see:- Newton’s “Axioms or Laws of Motion” starting on page 19 of volume 1 of the 1729 translation Archived 28 September 2015 at the Wayback Machine. of the Principia; Section 242, Newton’s laws of motion Archived 22 March 2015 at the Wayback Machine. In Thomson, W (Lord Kelvin), and Tait, P G, (1867), Treatise on natural philosophy, volume 1; and Benjamin Crowell (2000), Newtonian Physics.
For speed, velocity and acceleration: www.mytutor.co.uk/answers/8544/GCSE/Physics/What-is-the-difference-between-acceleration-speed-and-velocityFor eltromagnitism:
Ravaioli, Fawwaz T. Ulaby, Eric Michielssen, Umberto (2010). Fundamentals of applied electromagnetics (6th ed.). Boston: Prentice Hall. p. 13.
Whelan, P. M.; Hodgeson, M. J. (1978). Essential Principles of Physics (2nd ed.). John Murray
Clausius, Rudolf (1850). On the Motive Power of Heat, and on the Laws which can be deduced from it for the ‘Theory of Heat’. Poggendorff’s Annalen der Physik, LXXIX (Dover Reprint).