Calculus- part 1: Introduction

Calculus is the mathematical study of continuous change. Originally called infinitesimal calculus, it has two major branches, differential calculus and integral calculus.

Differential Calculus

The derivative of a function at a chosen input value describes the rate of change of the function near that input value. The process of finding a derivative is called differentiation. Geometrically, the derivative at a point is the slope of the tangent to a curve. The derivative of a function f(x) with respect to x is 

The above expression is also called as fundamental theorem of differentiation.

A partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant. The del symbol is used to denote the partial derivative. The chain rule is used to find the derivative of the multivariable function.

Integral Calculus

The concept of the integral was first started to calculate the area under a curve. The area under a curve can be thought of as a sum of rectangles of infinitesimal width. The integral of a function f(x) with respect to a variable x on an interval [a, b] is written as

If limits are specified, the integral is called a definite integral. When the limits are omitted, the integral is called an indefinite integral. There is no fundamental theorem of integration unlike differentiation.

Now let's look at the fundamental theorem of calculus which relates these two branches of calculus i.e. differential calculus and integral calculus. It states that the integral of f over an interval is equal to the antiderivative of f (a function whose derivative would be f). This result was kind of surprising and nonintuitive that somehow the problem of finding areas under a curve could be solved with differentials. 

Proof : For a given function f, define the function F(x) as

Differentiating on both sides






Integral from x to x+Δx is just the area under f in that interval. For a very small value of Δx, we can assume f(x) remains constant. Thus this area is equal to f(x)Δx.




Infinitesimal calculus was developed independently in the late 17th century by Isaac Newton and Gottfried Wilhelm Leibniz. Today, calculus has widespread roots in every branch of physics and engineering.

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Calculating atomic mass, A practical approach

Dalton published his first table of relative atomic weights containing six elements (hydrogen, oxygen, nitrogen, carbon, sulfur and phosphorus), relative to the weight of an hydrogen atom. Since these were only relative weights, they do not have a unit of weight attached to them.

The basic approach for measuring mass of an hydrogen atom is discussed below. The amount of electricity passed through the electrolyte is directly proportional to the mass of any substance deposited or liberated at an electrode, according to Faraday’s law of electrolysis.

        m = Z q ;     m - mass,  q - charge,  Z - constant of proportionality

        

The constant 'Z' can be experimentally determined by taking ratio of hydrogen liberated at electrode and the charge transferred q = I × t . It is approximately equal to 96500 gm/C. 

So the amount of charge required for depositing one hydrogen atom is same as the charge of an H+ atom that is q = 1.6 × 10^-19. Thus the mass of one hydrogen atom liberated at an electrode should be simply Z times q.  

        Atomic mass =  96500  × 1.6 ×10^-19     = 1.6 ×10^-27 kg

        

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What is Centrifugal Force ?

Centrifugal is not a real force. It's just the equivalent of the force needed to be applied to overcome inertia as per Newton's second law. When a body is moving in a circular path with constant speed, its velocity is constant in magnitude but not in direction. Thus the rate of change of velocity is not zero. As the body keeps changing direction and has non-zero acceleration, there must be some force acting on it.

Let's assume a body of mass m moving on a circular path of radius r. The body moves by a small angle dθ in time dt. Distance traveled by the body during this time is given by the product of the radius of the circle and the angle subtended at the center that is rdθ. Therefore dt can be written as

The angle between two lines is the same as the angle between their perpendiculars. Therefore v1 and v2 are also having angle dθ between them. Taking the x-axis along v1 and the y-axis perpendicular to v1 and radially outwards

Change in velocity is difference of final and initial

Centrifugal force is oriented along the negative y-axis as per the assumed coordinate system which is the direction towards the center. So whenever the body is revolving in a circular motion with constant speed, a centripetal force towards the center is required to provide for the inertial force. Centrifugal is not one of the four fundamental forces or derived from them unlike centripetal force rather it is the result of newton's second law.

As a body in circular motion has acceleration towards the center, the person inside the body experiences a pseudo force in opposite direction i.e. radially outward. It is just like a person inside a lift moving upwards experiences a downward pull. This phenomenon leads to the common misconception that centrifugal force is something special force acting radially outward.

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Illusion of Force and Mass

Let us start with the famous Newton’s second law of motion.

F = ma

   where

   F- net force acting on the body

   m- mass of the body

   a- acceleration of the body

What is the mass of a body? Generally, mass is attributed to the amount of matter in a body. How do we measure the mass of any body? Due to the unique property of gravitational force, on Earth’s surface weight (mg) of any body is proportional to mass. Dividing weight by constant g, the mass of a body can be obtained. While from the second law of Newton, we calculate force given the mass and acceleration, for measuring mass itself it is required to use some kind of force. This makes equation F=ma look like a cyclic illusion.

Even though we cannot measure mass directly without using some force, there is a way to verify the above equation. Let's say first we take a solid iron block and assume its mass to be m. We apply force F on it which results in the acceleration of the block. Now we brake the block into two halves. Assuming the mass of each resulting sub-block is m/2. We apply the same F force on the resulting block of mass m/2. Then we break this block into another two. Applied force F on the block of mass m/4. The acceleration for m/2 and m/4 should be 2a and 4a for the same force. Thus we can verify Newton's second equation by the method of cutting a standard block into pieces. Here it was assumed that mass is proportional to the volume of the body, as the block that is broken into two halves must have twice the matter than every two broken parts.

The important thing to note about equation F=ma is that force as such does not need to be defined by mass. There are four fundamental forces in nature 1. strong nuclear 2. weak nuclear 3. Electromagnetic 4. Gravitational. Force acting can be any one of them or derived from them. The consequence of this force on the body is acceleration which is related by the above equation. Where mass m is the intrinsic property of the body which decides how much acceleration will be generated for the given force. As seen in the previous example of a block, it is related to the amount of matter in a body but as such without force, there is no way to define mass.

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Faraday's law of Electromagnetic Induction

Michael Faraday, who became one of the greatest scientists of the 19th century, began his career as a chemist. He worked as an assistant to famous chemist Humphry Davy. Faraday had no formal education beyond basic reading, writing and math, and he never went to college. 

At the age of 14, he became an apprentice to a local bookbinder and bookseller. During his seven-year apprenticeship, Faraday read many books and developed interest in science. His main discoveries include the principles underlying electromagnetic induction, diamagnetism and electrolysis and the invention of electric motor and dynamo.

Faraday's Law states that the magnitude of emf induced in the coil is equal to the rate of change of flux that linkages with the coil. The flux linkage of the coil is the product of the number of turns in the coil and flux associated with the coil.

Faraday's law can be understood in-depth from Maxwell's equation

        

The above equation means If there is time-varying magnetic flux passing through any arbitrary loop drawn in space, then there is an electric field established such that the line integral of electric field E on a closed loop is equal to the rate of change of flux.

Note here that it is not required for the conductor or coil to be present to establish the electric field and hence emf. But generally, no current flows due to the high resistance of air.

Let's say we place a circular copper coil in a time-varying magnetic field, then free electrons in the copper will start moving in the direction of the electric field created as stated above. The current flowing, in turn, produces its own magnetic field and tries to effectively cancel out variations in original magnetic field. This is in accordance with Lenz's Law.

If     

            ...............................................(1)

            .......................................................(2)

Equating 1 and 2, we get

          

      

Suppose one turn coil replaced with N turn coil, then induced emf is simply N times that of single turn emf. This is because for each turn    and so for N turns in series ,the emf will add up.

        

Here N being multiplied with the rate of change of flux is a very critical concept in Faraday's law. It is essential that to apply Maxwell's equation loop must be closed since phi is flux passing through a closed loop. A turn is almost circle, but it is not a closed-loop, though, as said before, it is not required the conductor to be present; hence we take an arbitrary circle in space that has the same radius as the turn and then electric field E obtained is same as single turn coil. It is as if the circles of the electric field are already present in space, and we are just placing a coil there.

        

        

         

Since N is a constant  

        

        

Psi is called flux linkage, and instead of adding emf of all turns in series, it is dealt with by taking emf induced equal to rate change of flux linkage.

Faraday's law explains the working of electrical elements like Inductor and Transformer and is the basis for converting mechanical motion into electrical energy in all electric generators. 

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What exactly is Archimedes' principle

Archimedes was a Greek mathematician and scientist. He was born in the Greece city-state of Syracuse in 287 BC. A large part of Archimedes' work in engineering probably arose from fulfilling the needs of his home city of Syracuse. Archimedes' inventions include a proof of the principle of the lever, the widespread use of the concept of center of gravity, and the explanation of the law of buoyancy. He is also credited with designing innovative machines, such as his screw pump, compound pulleys, and defensive war machines to protect his native Syracuse from invasion.

If you put stone into a water container, volume of water displayed by stone is equal to volume of stone itself. When students are taught about Archimedes principle in class 8th or 9th they sometimes mistaken only above part as Archimedes principle but there is more to it. Lets look at more exact statement of the principle

"The upward buoyant force that is exerted on a body immersed in a fluid, is equal to the weight of the fluid that the body displaces".

Consider an experiment in which a stone of mass M attached to a sting is dropped into water. Let volume of stone be V which is also equal to volume of water displaced, density of stone ρ_s and that of water is ρ_w. Then according to Archimedes principle, 

Upward force on stone also called as buoyant force = weight of water displaced = ρ_wVg, 

while downward force on stone = Gravitational force = Mg = ρ_sVg.

Therefore net downward force on stone F = ρ_sVg - ρ_wVg = (ρ_s - ρ_w)Vg. 

This results into three cases depending on relative value between density of stone i.e. 

If ρ_s > ρ_w stone will sink into the water since F is positive. 

If ρ_s = ρ_w  stone will neither sink nor float since F=0. 

If ρ_s < ρ_w  stone will float into the water since F is negative

Actually, the origin of buoyant force can be understood as when stone wasn't there, water in that portion was prevented from falling down due to gravity by water below it . So to maintain the portion in equilibrium upward force equal to  ρ_wVg was always present and even after water was replaced by stone, there isn't any different treatment to it i.e. same upward force was applied on the stone.

Archimedes principle is useful measuring volume and density of irregular objects. It describes working of large ships and submarines.

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