Category: Physics Notes PPT

The equilibrium constant, k, is a number that describes the relationship between the number of products and reactants present at equilibrium in a reversible chemical reaction at a given temperature.

K_p and K_c are equilibrium constants of ideal gas mixtures considered under reversible reactions. K_p is an equilibrium constant written with respect to the atmospheric pressure and the Kc is the equilibrium constant used with respect to the concentrations expressed in molarity. The K_p K_c relation can be derived by understanding what are K_p and K_c. 

Let’s consider the general equilibrium equation:

A + B  ⇌ C + D

According to the law of mass action,

The rate at which A reacts ∝ [A]

The rate at which B reacts ∝ [B]

∴ The rate at which A and B react together ∝ [A][B]

So, the rate of the forward reaction = kf [A][B]

Where kf is the velocity constant for the forward reaction.

Now, the rate at which C and D react together ∝ [C][D] 

So, the rate of the backward reaction = k_b [C][D] 

k_b = velocity constant for the backward reaction.

At equilibrium, the rate of the forward reaction = rate of the backward reaction.

  kf [A][B] = kb [C][D]

  [frac{[C][D]}{[A][B]}=frac{kf}{kb}]

At constant temperature k_f and k_b are constant, therefore,[frac{kf}{kb}]=k is also constant at a constant temperature.

Here, k is called the Equilibrium constant.

Equilibrium Constant kc

Let us consider a general reversible reaction:

aA + bB ⇌ uU + vV

Applying the law of mass action here:

[frac{[U]^{u}[V]^{v}}{[A]^{a}[B]^{b}}]= k or kc

In terms of expression of concentrations, k is written as kc.

This mathematical expression is called the law of chemical equilibrium.

Equilibrium Constant Definition

At a constant temperature, the equilibrium constant is the ratio of the product of the molar concentrations of the products, each raised to the power equal to its stoichiometric coefficient and the product of molar concentrations of the reactants, each raised to the power equal to its stoichiometric coefficient.

Do you know how to find equilibrium constant k_p for the reaction?

Let’s derive the equilibrium constant formula for gas-phase reactions:

The Equilibrium Constant k_p for the Reaction

When both reactants and products are in gaseous states, then we express equilibrium constant either in terms of concentration in moles per liter or partial pressures of the reactants and the products. 

Derivation:

The relation between Kp and Kc is given by the following simple derivation. To derive the relation between Kp and Kc, consider the following reversible reaction:

‘a’ mole of reactant A is reacted with ‘b’ mole of reactant B to give ‘c’ moles of product C and ‘d’ moles of product D, 

aA + bB ⇌ cC + dD

Where a,b,c, and d are the Stoichiometric coefficients of reactants A, B and products C, D.

What is kc? 

kc is the equilibrium constant for a reversible reaction and it is given by,

[kc=frac{C^{c}.D^{d}}{A^{a}.B^{b}}]

Where,

C – The molar concentration of product ‘C’

D – The molar concentration of product ‘D’

A – The molar concentration of reactant ‘A’

B – The molar concentration of reactant ‘B’

Where,

PC – Partial pressure of product ‘C’

PD – Partial pressure of product ‘D’

PA – Partial pressure of reactant ‘A’

PB – Partial pressure of reactant ‘B’

Similarly,  Kp is the equilibrium constant in terms of atmospheric pressure and is given by the expression:

                aA + bB ⇌ cC+ dD

 Then, equilibrium constant formula for 

Where pv, px, py, and pz are the partial pressures of V, X, Y, and Z, respectively.

The partial pressures are taken in the following units:

1. Atm

2. Bar

3. Pascal

Relation between  Kp and Kc

To derive a relation between  Kp and Kc, consider the ideal gas equation,

PV = nRT

Where,

P – Pressure of the ideal gas

V – Volume of the ideal gas

n – Number of moles

R – Universal gas constant

T – Temperature

On rearranging the above equation for P,

P = nRT/V…………..(3)

We know that the ratio number of moles per unit volume is the molar concentration of the substance, hence we can write the pressure equation as:

P = molar concentration RT ………………..(4)

Therefore the partial pressures of A, B, C, and D can be calculated by using equation (4):

⇒ PA = A RT

⇒ PB = B RT

⇒ PC = C RT

⇒ PD = D RT

Let us consider a general reversible reaction equation:

                         aA + bB ⇌ cC + dD

Substituting the above values in equation (a), and simplify:

[kp=frac{C^{c}.D^{d}RT^{c+d}}{A^{a}.B^{b}.RT^{a+b}}]

Also, we may write equilibrium constant  (kc)  in terms of molar concentrations as;

So, from eq 

[kp=frac{C^{c}.D^{d}RT^{c+d}}{A^{a}.B^{b}.RT^{a+b}}]

[kp=kc ast RT^{(c+d)-(a+b)}]

Where,

c + d – Number of moles of product = np

a + b – Number of moles of reactant = nr

Therefore, 

(c + d) – (a + b) = np – nr = Δng

Thus we get the relation between Kp and Kc,

kp = kc [(RT)^{Delta n}]

Where,

Δng – Change in gaseous moles of reactant and the product.

This is the required expression that gives the relation between the two equilibrium constants. The relation between Kp and Kc Pdf can be downloaded. Depending on the change in the number of moles of gas molecules, Kp and Kc relation will be changing.

In other terms, we have kc in molar concentration in the following manner:

[kc=frac{CC^{c}.CD^{d}}{CA^{a}.CB^{b}.}]….(b) 

Ca, Cb, Cc, and Cd express the molar concentrations of A, B, C, and D, respectively. 

If we consider the gas as an ideal, then we can apply the ideal gas equation, that is:

pV = nRT or p = [frac{n}{v}]RT = CRT

∵ [frac{n}{v}] =[frac{text{Number of moles}}{text{Litre}}]= C (Molar concentration)

∴ For the gases V, X, Y, and Z, we may write the equation (1) and (2) as;

            pa  = CaRT …(a)

            pb  = CbRT …(b)

            pc  = CcRT …(c)

            pd  = CdRT …(d)

Now, putting values of equations (a), (b), (c), and (d) in equation (2):

             kp  = [frac{(CcRT)^{c}.(CdRT)^{d}}{(CaRT)^{a}.(CbRT)^{a}}]

                  = [frac{CC^{c}.CD^{d}.(RT)^{c+d}}{CA^{a}.CB^{b}.(RT)^{a+b}}]

                  = [frac{CC^{c}.CD^{d}}{CA^{a}.CB^{b}}(RT)^{(c+d)-(a+b)}]  

                  =  kc [(RT)^{Delta n}]

 Where kc =[frac{CC^{c}.CD^{d}}{CA^{a}.CB^{b}}]from equation (2), and

 Δn = (c + d) – (a + b) 

       = No. of moles of products – No. of moles of reactants

       = Change in the number of moles

Hence,             

Where R is the Universal Gas Constant whose value is 0.821 liter-atm per degree kelvin-mole, and

T = Temperature in degree Kelvin (°K).   

Here, Δng – Change in gaseous moles of reactant and the product.

This is the required expression that gives the relation between the two equilibrium constants. The relation between kp and kc Pdf can be downloaded. Depending on the change in the number of moles of gas molecules, Kp and Kc relation will be changing.

Case-1:

If Δng = 0, i.e., if the change in the number of moles gas molecules in the equation is zero.

Then Kp = Kc

Case-2:

If the change in the number of moles of gas molecules is positive, i.e., if Δng > 0 then,

Kp > Kc

Case-3:

If the change in the number of moles of gas molecules is negative, i.e., if Δng < 0 then,

Kp < Kc

Equilibrium Constant Units     

For the general reaction: aA + bB ⇌ cC + dD               

kc =[frac{CC^{c}.CD^{d}}{CA^{a}.CB^{b}.}] = [frac{(MolL^{-1})^{c+d}}{Mol L^{-1})^{a+b}}] = [(Mol L^{-1})^{(c+d)-(a+b)}=(Mol L^{-1})^{Delta n}]

So, the unit of kc is[Mol L^{-1}], and

kp = [frac{PC^{c}.PD^{d}}{PA^{a}.CP}]=  [frac{(atm)^{(c+d)}}{(atm)^{(a+b)}}]= [(atm or bar)^{(c+d)-(a+d)} or (atm or bar)^{Delta n}]

So, the unit of kp = atm or bar.

Application of Equilibrium Constant

One of the applications of the equilibrium constant is to predict the extent of reaction. 

The magnitude of the equilibrium constant gives an idea of the relative amounts of the reactants and products.

For example, consider the following reversible equation and hence calculate Kp and Kc and derive the relationship between Kp and Kc:

                H2 (g) + Br2 (g) ⇌ 2 HBr(g) (kp = 3 x 1019)

                H2 (g) + Cl2 (g) ⇌ 2 HCl(g) (kp = 5 x 1029)

Here, values of kp are very high, i.e., reactions go almost to completion. 

Another Example:

  H2  + I2   ⇌ 2 HI

Solution:

Given the reversible equation,

  H2  + I2   ⇌ 2 HI

The change in the number of moles of gas molecules for the given equation is,

⇒ Δn = number of moles of product – number of moles of reactant

⇒ Δn = 2 – 2 = 0

Therefore, Kp = Kc

Then, Kp and Kc of the equation is calculated as follows,

 [kc= frac{HI^{2}}{H^{2}I^{2}}]

Solved Examples on Kp and Kc

Example 1: For the reaction,

N2O4(g) ⇌ 2 NO2(g) 

The concentration of the equilibrium mixture at 293 K of N2O4 is 5 x 10-8mol/L, and of NO2 is 2 x 10-6mol/L. Find the value of the equilibrium constant.

Applying the formula, k =[frac{[NO]^{2}}{[N2O4]}]

Taking the concentrations w.r.t. standard state concentration of 1 mol/L:

k =[frac{(2×10^{-6})^{2}}{5×10^{-8}}=8×10^{-5}]

Example 2: For the reaction,

N2 (g) + 3H2 (g) ⇌ 2 NH3(g)

If pN2 = 0.30 atm, pH2 =  0.20 atm, and pNH3 = 0.40 atm, then what is the value of kp?

Using the formula, kp =[frac{pNH3^{2}}{pN2.pN2^{2}}]

=[frac{(0.4)^{2}}{(0.3).(0.2)^{2}}]= 13.3 atm

In Physics, every physical entity can be measured in different ways. Every unit can be related to one another by performing unit conversions without violating the laws of physics and laws of nature. 

 

Newton is the SI unit of force and kg is the unit of mass. According to Newton’s Second law of motion, force is directly proportional to the mass of the object on which force has been exerted. Thus we can say Newton and Kg are also directly proportional to each other, thus if we encounter any change in the unit of force in Newton it will result in a change in the unit of mass in Kg keeping the acceleration constant.

 

To arrive at a mathematical description for the relation between Kg and Newton, let us discuss the definition of Kg and Newton respectively. In this article, we try to explain to students the concept with great simplicity and clarity making the students grasp the knowledge with ease. 

 

Table of Content – 

• Introduction

• Meaning of KG

• Meaning of Newton

• Derivation

• Examples 

• Learning from the topic 

• Benefits of referring the notes 

• Frequently asked questions 

 

What is Kg?

There are seven fundamental units in physics, Kg is one of them. (Fundamental units are the units that are independent quantities, all other units are derived from them). The Abbreviation of Kilogram is Kg.

 

1 Kilogram is nearly equal to 1000grams. The kilogram is one of the basic units of metric systems.

 

What is Newton?

Newton is the SI unit of Force. It is defined as the force required to accelerate an object of the mass of 1 kilogram (1kg) by 1m/s² in the direction of applied force. In the CGS system, 1N is equal to 10⁵ Dyne. Dyne is the unit of Force in the CGS system.

 

Derivation

Mathematically we can describe one Newton by using Newton’s second law of motion,

I.e.,

 

⇒ F = ma

 

⇒ 1 Newton = 1Kg x 1m/s²

 

From the above expression, Newton is directly proportional to Kg. Therefore,

• If the object under consideration is having negligible mass or considerably less mass then the force required in Newton will also be very less.

• If the object an action-heavy, then the force required will also be more.

1N is Equal to How Many Kg?

We can not convert Kg to Newton, because Newton is a unit of Force whereas Kg is the unit of mass respectively. Conversion of units can be done for two identical scales but not for two different physical scales. So, we can give a relation for the two units, saying conversion of them will be the wrong term.  

 

When an object is dropped from a certain height above the ground level, it will experience a force purely due to the acceleration due to gravity. While free fall the object will experience a force and the force experienced is known as Weight of the body/object and mathematically is given by:

 

W = mg ……..(1)

 

Now we can give a relation between newton and kg by analyzing how many Kg in one Newton. Let us assume that a 1Kg mass is dropped from a height above ground level with an initial velocity zero then the force experienced by the object is,

 

W = 1Kg x 9.81 m/s² = 9.81 N

 

From the discussion of the relation between Kg and Newton, we have concluded that they are directly proportional to each other. Therefore, we write:

 

1Kg=9.81N

 

Therefore, 1 newton is equal to how much Kg or 1 kg how many newtons? The answer is 9.81N.

 

Similarly, If the question demands for 1 kg wt is equal to how many newtons? (Which is in terms of gravitational force) Then also the answer is either 9.8N or 9.81N or 10N because the value of acceleration due to gravity is considered depending upon the convenience.

 

Example

1. The mass in Kg of an object that Weighs about 40N.

Ans: Given that weight of the object is 40N.

We are asked to find the mass of the object.

From the relation between Kg and Newton we have,

⇒ 1Kg = 9.81N

⇒ 1N = 0.102Kg

Thus,

⇒ 40 N = 40 x 0.102Kg = 4.077Kg

Therefore, the mass in kg of an object that weighs about 40N is 4.077.

An alternative method for calculating the mass of an object whose weight is given is as follows:

We know that,

⇒ 1N = 1Kgm/s²

Therefore,

⇒40N = 40Kgm/s²

We want to convert 40N into Kg, to do that divide the above expression by acceleration due to gravity, i.e., divide by 9.81 m/s²

⇒ 40N = 40Kgms−2/9.81 ms⁻²

⇒ 40N = 4.077kg

This is the required answer.

 

2. How many Newtons is 5Kg?

Ans: We know that 1kg = 9.81N, then:

⇒ 5Kg = 5 x 9.81 N = 49.05N

therefore,49.05N is 5Kg.

 

Learning From the Topic – 

• Every object has a unit of measurement, which measure in different ways and forms 

• Kg and Newton are also forms of units of measurement 

• Kg is the unit of a mass where kg equals 1000 grams of mass of an object

• Newton is the unit to measure the force required to accelerate an object of 1kg of mass.

• Newton’s second law of motion is used to derive one newton. The second law says the rate of change of linear momentum of a body is directly proportional to the external force applied to the body in the direction of applied force. 

• Therefore, 1 Newton = 1Kg x 1m/s²

Read the full article to understand more about the concepts and try t
o solve a few practical questions to evaluate your level of understanding of the topic. 

 

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The term “resistor” refers to a device that acts as a two-terminal passive electrical component that is used to limit or regulate the flow of electric current in electrical circuits. And it also allows us to introduce a controlled amount of resistance into an electrical circuit. The most important and commonly used components in an electronic circuit are resistors.

A resistor’s main job is to reduce current flow and lower voltage in a specific section of the circuit. It’s made up of copper wires that are wrapped around a ceramic rod and coated with insulating paint.

The basic idea is known to all about how electricity flows through an electronic circuit. Here, two categories can be identified which are conductors and insulators. Insulators do not allow the flow of electrons, but the conductor does. However, the resistor determines the amount of electricity that is allowed to pass through them. The total voltage passes through when it is passed through a conductor like the metal; by introducing the resistors, the amount of voltage and current can be controlled. 

The ease at which the electrons will allow the electricity to flow through it is known as resistance.

An insulator has better resistance than the conductor, and the term resistance is defined as the electrical quantity used by the resistor to control the flow of electrons.

 

What is Resistance?

Based on Ohm’s law named after German Physicist Georg Simon Ohm, the resistance is defined as follows:

As per the Ohm’s Law, the voltage V across a resistor is directly proportional to the current I flowing through it. Here, the resistance R is constantly proportional.

Therefore, V = I [times] R

 

Resistor Unit

The SI unit of resistance is known as Ohm Ω. Kiloohms KΩ, megaohms MΩ, milliohm, and so on are known as the higher multiple and sub-multiple values of Ohm.

The voltage required for creating 1 ampere of current to flow through the circuit is known as the resistance. For example, if we have to create 1 ampere of current flow through a circuit by 100 volts, then the resistance is 100 ohms.

 

Resistor Symbol

The resistor symbol is given below.

 

Each resistor has two terminals and one connector. We’ll look at the three different types of symbols used to represent a resistor.

Each of the lines coming from the squiggle is the resistor’s terminals (or rectangle). These are the wires that connect the circuit to the rest of the components. Both a resistance value and a name are commonly added to resistor circuit symbols. The ohms value is obviously important for both analysing and actually constructing the circuit.

 

How does a Resistor Work?

Water flowing through the pipe can be used as an example to explain the working of the resistor. Consider a pipe through which the water flows. Now, as the diameter of the pipe is reduced, the flow of the water will be reduced. Further, as the pressure is increased, the force of the water is increased, and energy is dispersed as heat. With this example, the force applied to the water is similar to the current flowing through the resistance. The voltage can resemble the pressure applied.

 

Working Principle of Resistor

The resistor absorbs the electrical energy in the process where it acts as a hindrance to the flow of electricity by reducing the voltage, and it is dissipated as heat. In today’s world of electronic circuits, the heat dissipation is typically a fraction of a watt.

Ohm’s law states that if I is the current flowing through the resistor in amperes, and R is the resistance in ohms, then V is the voltage drop that is imposed by the resistor (it is the electrical potential difference between the two contacts that are attached.).

[V = frac{I}{R}]

Another way of saying this is that the 1Ω resistor will allow a current of 1 amp when there is a capacity difference between the ends of the resistor of 1 volt. 

If P is the power in watts dissipated by the resistor, in a DC circuit:

P = V [times] I

By substitution of Ohm’s law, we can express power (watts) in terms of current and resistance:

[P = frac{I^{2}}{R} ]

We can also express power (watts) in terms of voltage and resistance:

[P = V^2 times R]

These alternative equations can be used when you do not know the value of the voltage drop or the current, respectively.

Approximately similar relationships exist when using alternating current, although the power will be a more complex function of the resistor.

 

Resistor Series and Parallel Circuits

There are cases in which an electrical circuit might have two or more resistors. They can be connected in series and parallel ways.

The resistors, when connected in the series path, are known as a series connection, and the current flowing through them will be the same. The sum of the voltage across each resistor will be equal to the voltage across the resistors. Here is a diagram of resistors connected in series. In a series connection, the three resistors [R_{1}, R_{2}] and [R_{3}] and total resistance [R_{total}] is given by:

[R_{total} = R_{1}  +  R_{2}  + R_{3} ]

The series in which the resistors are connected in a parallel is known as a parallel connection. Here, the voltage applied across each component remains the same. The sum of the currents across each resistor is equal to the current across the series.

The below diagram shows the parallel-series connection of resistors.

Here, the three resistors named [R_{1}, R_{2}] and [R_{3}] are connected.

The total resistance [R_{total}] is given by

[frac{1}{R_{1}} + frac{1}{R_{2}} + frac{1}{R_{3}} = frac{1}{R_{total}}]

Therefore, [R_{total} = frac{(R_{1} times R_{2} times R_{3})}{(R_{1} + R_{2} + R_{3})} ]

 

Power Dissipated in a Resistor

The below equation will give you the value of the power dissipation through a resistor.

Power [P = I^{2} R = V I = V^{2} ] 

The first equation was obtained from Joule’s first law, while the other two were derived from Ohm’s law.

Types of Resistors

Resistors are available in a wide range of shapes and sizes. Common varieties that are offered are through-hole and surface mount. A static resistor, a normal resistor, a customised resistor, or a pack of variable resistors are all examples of resistors.

The following are the two basic types of resistors:

Linear resistors have values that fluctuate when the temperature and voltage applied to them change. Linear resistors are divided into two categories:

Fixed resistors are those that have a fixed value that cannot be changed. The following are the several types of fixed resistors:

• Resistors with a carbon content

• Wire-wound resistors are a form of wound resistor

• Thin-film resistors 

• Resistors with a thick film

Ohm’s law does not apply to resistor values, which alter with temperature and applied voltage. The following are the numerous types of non-linear resistors:

Resistor Applications

Resistors are used in the following ways:

• In shunt with ampere metre applications where balanced current regulation, high sensitivity, and precise measurement are required, wire-wrapped resistors are used.

• Photo resistors are used in flame detectors, burglar alarms, and photographic devices, among other things.

• Resistors are used to control temperature and voltmeter readings.

• Digital multimeters, amplifiers, telecommunications, and oscillators all employ resistors.

• Modulators, demodulators, and transmitters all use them.

 

Summary

During the study of electricity, you discover the materials that are used, which are categorized into two basic categories, namely conductors and insulators. The substance such as metal through which the electricity flows is known as a conductor. Besides, the material like plastic and wood through which the electricity doesn’t flow is known as an insulator. But it is not as simple. Electricity can be conducted through a substance if it can flow significant voltage across it: even air, which is considered as an insulator, can become a conductor when powerful voltage builds up in the clouds, which causes the lightning to happen.

 

If you want to understand the ease with which electricity flows, it is better to talk about the resistance rather than discussing insulators and conductors. As compared to the conductor, the insulator has a much higher resistance. 

Sir Isaac Newton once wondered why apples on the trees fall to the ground. Trying to find an answer to this question, he proposed the laws of gravity in 1687 and gave an answer to his own question. But if gravity acts downwards, why do things which are pushed forwards on a surface eventually stop? Why do the wheels of a cart that is pushed forward eventually stop? Why do we need to maintain a pushing force when we want to push something to a spot some distance away? Attempts to answer such questions were made by Leonardo da Vinci as early as 1493 when he documented the classic laws of sliding friction in his unpublished notebook. These laws were rediscovered in 1699 by Guillaume Amontons. Friction is the force that resists motion between two surfaces that are sliding against each other. There are various kinds of friction. And one among them concerns bodies that are in motion by rolling across a surface.

 If you kick a football, it will roll across the ground for a certain distance before coming to a stop. We can infer by this that the energy you supplied to the football by kicking it dissipated after it rolled across the ground for a certain distance. We can conclude that there is a certain resistance against the ball as it rolls across the ground which drains its energy eventually causing it to stop due to the lack of energy. This resistive force that the ground applies on the rolling football is rolling friction.

Rolling friction is the resistive force offered by any surface which opposes the rolling motion of any object that rolls over it, thus causing it to slow down and eventually stop. Rolling friction occurs when a spherical or round object rolls across a surface. Rolling friction is also sometimes called rolling drag or rolling resistance.

In this chapter, you will learn the following concepts – 

• Rolling friction – An introduction

• Differences Between Rolling and Sliding Friction:

• Formula of Rolling Friction

• Coefficients of Rolling Friction

• Factors Influencing Rolling Friction

• Frequently asked question 

 

Differences Between Rolling and Sliding Friction:

When a spherical or round body rolls across the surface of across a surface, the resistance in motion which arises is rolling friction. On the other hand, when an object with a flat side slides across a surface, the resistance in motion which arises is sliding friction.

It is much easier to roll an object than to slide the same object. When a spherical or round object rolls, it has only a part of its surface in contact with the ground, thus there is less resistance offered by the ground towards it. But when an object with a flat side slides across the surface, all of that side is in contact with the ground, and it bears maximum resistance from the ground. A simple example for this difference is the gas cylinder; it is difficult to slide a gas cylinder across the floor, but it is easier to tilt the cylinder to the side and roll it across the floor. Not only it is easier to roll an object over sliding it, but it is also faster and more convenient. Thus, objects like stands, furniture, and bigger decorations have tiny wheels under them sometimes; this makes it easier to move them around. Other than this, the coefficient of rolling friction is smaller than the coefficient of sliding friction under the same conditions.

 

The Formula to Calculate Rolling Friction:

The general equation to calculate rolling friction is,

Fr = μrN

where:

Fris the rolling friction, or the resistance towards rolling objects,

μris the coefficient of rolling friction,

N is the normal reaction on the object 

μr, which is the coefficient of rolling frictions can be defined as the ratio of the force of the rolling friction to the total weight of the object.

Rolling resistance can also be expressed as,

Fr = μrW  

where:

Fris the rolling friction, or the resistance towards rolling objects,

μr is the coefficient of rolling friction,

W is the weight of the object,

Following are some typical coefficients of rolling friction we come across in our everyday life.

Some Typical Coefficients of Rolling Friction:

 

Factors that Influence Rolling Friction:

There are many factors that subtly influence rolling friction, like the shape of the wheel, type of surfaces, speed of the wheel and pressure on the wheel.

Among these many tiny factors, the factors that directly influence the friction and inhibit motion are

1. Elastic Deformations

2. Surface Irregularities

3. Molecular Friction

1. Elastic Deformations: Materials like rubbers are popularly used to make tires deform when pressure is applied to them. But we may not observe that even hard materials like asphalt or concrete deform slightly
   when pressure is applied to them, by the weight of something like a car which goes over them. These deformations which are in contact are the major factors inhibiting motion. Material choosing is important precisely due to this reason. If the wheels of a car are made up of steel or iron which have lesser potential to deform, not considering the added weight, it becomes much more difficult to control the car. Rubber tires filled with air are used to have the best control of the car.

 

 

2. Surface Irregularities: The surface of the entire wheel and the surface of the ground come into contact when the wheel is spinning. But the road will never be completely even. The wheel too will not be perfectly even. There will be irregularities on both surfaces. This roughness of the surface is a reason for resistance. Gears and roller bearings are polished to prevent surface irregularities and reduce friction. On the other hand, grooves are added to tires to increase friction and thus give better control and braking.

 

 

3. Molecular Friction: Molecular friction is caused by the molecular attraction or molecular adhesion between the materials used to make the wheel or any other rolling object and the surface on which the object travels. This is the reason why the materials used to construct the wheel and surface is important. This molecular friction can be viewed as a kind of “stickiness” factor between the material of the wheel and the material of the surface on which the wheel travels. When some materials are pushed together, molecular force tries to keep them together and prevents them from being pulled apart. For example, if you roll a highly polished metal ball over materials like rubber in the form of a sheet, you can see that the metal ball does not really roll well as it slightly sticks to the rubber sheet. This can be thought of as a wheel with glue applied to it rolling across a surface. This applies only to specific materials.

Most of the quantities that we know are generally classified as either a scalar quantity or a vector quantity. There is a distinct difference between scalar and vector quantities. Scalar quantities are among those quantities where there is only magnitude, and no direction. Their results can be calculated directly. 

For vector quantities, magnitude and direction, both must be available. Hence, the result calculated will also be based on the direction. One can consider displacement, torque, momentum, acceleration, velocity, and force as a vector quantity. 

When it comes to calculating the resultant of vector quantities, then two types of vector product can arise. One is true scalar multiplication, which will produce a scalar product, and the other will be the vector multiplication where the product will be a vector only. 

In this article, we will discuss the scalar product in detail.

Scalar Product of Two Vectors

The Scalar product is also known as the Dot product, and it is calculated in the same manner as an algebraic operation. In a scalar product, as the name suggests, a scalar quantity is produced.

Whenever we try to find the scalar product of two vectors, it is calculated by taking a vector in the direction of the other and multiplying it with the magnitude of the first one. If direction and magnitude are missing, then the scalar product cannot be calculated for vector quantity.

To understand it in a better and detailed manner, let us take an example-

Consider an example of two vectors A and B. The dot product of both these quantities will be:-

[widehat{A}] . [widehat{B}] = ABcos𝜭

Here, θ is the angle between both the vectors.

For the above expression, the representation of a scalar product will be:-       

[widehat{A}] . [widehat{B}] = ABcos𝜭 = A(Bcos𝜭) = B(Acos𝜭)

We all know that here, for B onto A, the projection is Bcosα, and for A onto B, the projection is Acosα. 

Now, we can clearly define the scalar product as the product of both the components A and B, along with their magnitude and their direction. For the product of vector quantities, it is important to get the magnitude and direction both.

Commutative Law

Commutative law is related to the addition or subtraction of two numbers. This law is also applicable to scalar products of vectors. This property or law simply states that a finite addition or multiplication of two real numbers stays unaltered even after reordering of such numbers. This goes with the vectors also. The result of a scalar product remains unchanged even after the reordering of vectors while extracting their product. 

[widehat{A}] . [widehat{B}] = [widehat{B}] . [widehat{A}]

Distributive Law

The distributive law simply states that if a number is multiplied by a sum of numbers, the answer would be the same if such number would have been multiplied by these numbers individually and then added. This distributive law can also be applied to the scalar product of vectors. For better understanding, have a look at the example below-

[widehat{A}] . ( [widehat{B}] + [widehat{C}] ) = [widehat{A}] . [widehat{B}] + [widehat{A}] . [widehat{C}]

[widehat{A}] . λ [widehat{B}] = λ ([widehat{B}] . [widehat{A}])

Here, λ is the real number.

After understanding the commutative law and distributive law, we are ready to discuss the dot product of two vectors available in three-dimensional motion.

All of the three vectors should be represented in the form of unit vectors.

[widehat{A}] – Axi

[widehat{A}] = Axi + Ayj + Azk

[widehat{B}] = Bxi + Byj + Bzk

Here,

For X- Direction the unit vector is i

For Y- Direction the unit vector is j

For Z- Direction the unit vector is k

Now, when it comes to looking at the scalar product of all these two factors, it will be given by:-

 [widehat{A}] . [widehat{B}] = (Axi + Ayj + Azk) . (Bxi + Byj + Bzk)

[widehat{A}] . [widehat{B}] = AxBx + AyBy + AzBz

Here,

[widehat{i}] . [widehat{i}] = [widehat{j}] . [widehat{j}] = [widehat{k}] . [widehat{k}] = 1

[widehat{i}] . [widehat{j}] = [widehat{j}] . [widehat{k}] = [widehat{k}] . [widehat{i}] = 0

Solved Examples 

Question :- There is a force of F = (2i + 3j + 4k) and displacement is d = (4i + 2j + 3k), calculate the angle between both of them?

Answer:- We know, A.B = AxBx + AyBy + AzBz

Thus, F.d= Fxdx + Fydy + Fzdz 

= 2*4 + 3*2 + 4*3 

= 26 units

Alternatively,

F.d= F dcosθ

Now, F² = 2² + 3² + 4²

= √29 units

Similarly, d² = 4² + 2² + 3²

= √29 units

Thus, F d cosθ = 26 units.

Vector Quantity Definition

A vector quantity is a mathematical quantity that is defined by its magnitude and direction as two distinct qualities. The magnitude of the quantity with absolute value is represented here. In contrast, direction represents the north, east, south, west, north-east, and so on.

The vector quantity follows the triangle law of addition. A vector is represented by a vector quantity depicted by an arrow placed over or next to a symbol.

Difference between a Scalar and a Vector Quantity

A scalar quantity differs from a vector quantity when it comes to direction. Vectors have direction, whereas scalars do not. Because of this property, a scalar quantity is considered to be one-dimensional, whereas a vector quantity might be multi-dimensional. Let’s understand more about the differences between scalar quantity and vectors from the table below.

Important Differences Between Scalar Quantity and Vector Quantity

In terms of the scalar and vector difference, the following points are essential:

• The magnitude of a quantity is referred to as a scalar quantity. The vector quantity, on the other hand, considers both magnitude and direction to characterize its physical amount.

• Scalar quantities can describe one-dimensional numbers; for example, a speed of 35 km/h is a scalar quantity. Multidimensional values, such as temperature increases and decreases, can
  be stated using vector quantities.

• When only the magnitude changes, the scalar quantity changes; however, for the vector quantity, both the magnitude and direction must change.

• Scalar quantities conduct operations using standard algebra rules such as addition, multiplication, and subtraction, whereas vector quantities do operations using vector algebra rules.

A scalar quantity can also divide another scalar amount, however, two vector quantities cannot be divided.

Conclusion

aims at providing conceptual basis for topics they divulge in and Solutions to questions in an elaborate manner. You can find everything you’re looking for at ’s site, which is religiously designed by subject matter experts . These are in PDF forms and are easily downloadable for free. Students can go through these uniquely designed conceptual notes with Definitions and questions carefully and understand the tricks used to solve the sample questions. This will help them immensely in their examinations. 

The Seismograph

A seismograph, or seismometer is known as an instrument which is used to record and detect earthquakes. Generally we can say that it consists of a mass which is attached to a fixed base. Now during an earthquake the base moves and the mass does not. The motion of the base which is with respect to the mass is commonly said to be transformed into an electrical voltage. The electrical voltage which we are talking about is recorded on paper, and magnetic tape, or another recording medium. This record is said to be proportional to the motion of the seismometer mass relative to the earth.

But it can be mathematically also converted to a record of the absolute of the motion of the ground. The term or device seismograph generally refers to the seismometer and its recording device as a single unit.

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What is Seismograph

A seismometer is known as an instrument that responds to ground motions.

 Such as which is caused by earthquakes and the volcanic eruptions and explosions. The device that is seismometers are usually combined with a timing device and a recording device to form a seismograph. The output of such a device that is formally recorded on paper or we can say film.

Now recorded and processed digitally this is a seismogram. Such data which we have seen just above is used to locate and characterize earthquakes and to study the Earth’s internal structure as well.

The term seismograph is an instrument that generally makes a record of waves which is seismic and is caused by an explosion of the earthquake or other Earth-shaking phenomenon. The seismographs are said to be equipped with the electromagnetic sensors that generally translate the ground motions into charges which are electrical. Which are processed and generally recorded by the instruments’ that are digital or analog circuits. The terms seismograph and seismometer as well are often used interchangeably however we can say that  whereas both devices may detect and measure the waves which are seismic. Here we can say that only a seismograph possesses the capacity to record the phenomena. A record that is produced by a seismograph device on a display screen or the paper printout is known as a seismogram.

Although we can say that originally this was designed to locate natural earthquakes, the seismographs have many other uses as well such as petroleum exploration and investigation of planet earth’s crust and lower layers as well and monitoring of volcanic activity.

What is a Seismometer

An early instrument which was for seismic was known as the seismoscope made no time record of ground oscillations but simply indicated that shaking had occurred. A scholar who was from China named Zhang Heng. He invented such an instrument as early as 132 CE. It was said to be a cylindrical device in shape with eight dragon heads that were arranged around its upper circumference. That too with each ball in its mouth. Around the circumference were eight frogs which were each directly under a dragon head. When an earthquake occurred then at that time the balls were released from a dragon’s mouth which is probably by an internal pendulum that moved back and forth according to the direction of vibration. and along with that these were caught by a frog’s mouth which produced noise.

In 1855 an Italian scientist named Luigi Palmieri designed a seismograph that consisted of several tubes which were U-shaped and they were filled with mercury and oriented toward the different points of the compass. When the ground shook at that time the motion of the mercury made an electrical contact that stopped a clock and simultaneously it started a recording drum on which the motion of a float on the surface of mercury was registered. This device is thus said to indicate time of occurrence and the relative intensity and duration that is of the ground motion.

Seismograph Meaning

The word derives from the Greek which is denoted as: σεισμός, seismós. A shaking or quake that is from the verb σείω, seíō to shake and μέτρον métron that means to measure and was coined by sir David Milne-Home in 1841. This was to describe an instrument which is designed by Scottish physicist James David Forbes.

The instrument that is seismograph is another Greek term from seismós and γράφω, gráphō which means to draw. It is often used to mean seismometer though it is more applicable to the instruments which were older in which the measuring and recording of ground motion were combined. As compared to that of a modern system in which these functions are separated. Both types provide a record which is continuous of ground motion. This record which we are talking about distinguishes them from seismoscopes which merely indicate that motion has occurred. Perhaps we can say that with some simple measure of how large it was.

The technical discipline which is concerning such devices is known as seismometry, or a branch of seismology.

The concept that is of measuring the “shaking” of something means that the word “seismograph” might be used in a more general usage. For example we can say that a monitoring station that tracks changes in electromagnetic noise affecting amateur radio waves presents an rf seismograph. And Helioseismology studies the term “quakes” on the Sun.

The first seismometer was made in the country China during the 2nd century. The first Western description of the device comes from the physicist who was from French and priest Jean de Hautefeuille in 1703. The seismometer which was modern was developed in the late 19th century.

In December 2018, there was a seismometer which was deployed on the planet Mars by the InSight lander. It was the first time a seismometer was placed onto the surface of some other planet.

Today we can see that the most common recorder is a computer with an analog-to-digital converter. That is a disk drive and an internet connection for amateurs purpose of a PC with a sound card and associated software is adequate.