Momentum Vector

Introduction:

Measurement of motion of a body can be explained by momentum vector.

Definition of momentum vector:

Momentum vector is defined as the total motion contained in the body. Mathematically, momentum vector is equal to the product of mass of the body and its velocity.
P = m × v
Where m is the mass of the body and v be the velocity of the body.
Momentum vector is a vector quantity. The unit of momentum vector is kg m /s in MKS and g cm /s in CGS.

Principle of Conservation of Momentum Vector:

It states that if no external force is applied on a system, then the momentum of the system remains constant. In other words, if there is no external force applied on the system,
the initial momentum of the system will be equal to the final momentum of the system Consider a system of two bodies on which there is no external force acting on it. Because the system is isolated from the surroundings, so it interacts only due to their mutual interactions. Due to the mutual interaction, the momentum of the individual bodies may change but the total momentum of the system remains constant. If q₁ and q₂ be their individual momentum's, then
q₁ + q₂ = constant
For a system of n bodies, we can say that, q₁ + q₂ + q₃ +…..+ q_n = constant

Practical Application of Principle of Conservation of Momentum Vector:

When a bullet is fired from the gun, the gun recoils or gives a jerk in the shoulder in the backward direction. Let M be the mass of gun and m be the mass of the bullet. Initially both bullet and the gun are at rest. On firing the gun, suppose that the bullet moves with velocity v and the gun moves with velocity V. As we use the principle of conservation of momentum,
Total momentum of the gun and the bullet before firing = total momentum of the
         bullet and the gun after firing
0 = MV + mv
V = - mv / M
The negative sign shows that the gun will move in the opposite direction of bullet.

Angular Momentum for Rigid Body

Introduction:

The angular momentum definition of a particle is defined as the moment of linear momentum of the particle.
Let us consider a system of n particles of masses m₁,m₂,m₃..............m_n situated at distances of r₁,r₂,r₃,.......,r_n respectively from the axis of rotation .

Let v₁,v₂,v₃, ....be the linear velocities of the particles respectively, then the linear momentum of the first particle = m₁v₁

Definition of Angular Momentum for Rigid Body:

Since v₁ = r₁`omega`
Linear momentum of the first particle =m₁(r₁`omega` )
The moment of linear momentum of first particle = (m₁r₁`omega` )x r₁
Angular momentum of first particle = m₁r₁²`omega`
Similarly angular momentum of the second particle = m₂r₂²`omega`
and angular momentum of the third particle = m₃r₃²`omega` and so on.
The sum of moment of the linear momenta of all the particles of a rotating rigid body taken together about the axis of rotation is known as angular momentum of a rigid body.

Calculating the Angular Momentum for Rigid Body:

`:.` Angular momentum if the rotating rigid body = sum of the angular momenta of all the particles
`rArr` L = m₁r₁²`omega` +m₂r₂²`omega` +m₃r₃²`omega` +.............+m_nr_n²`omega`
`rArr` L = `omega` [ m₁r₁² +m₂r₂²+m₃r₃²+.....+m_nr_n²]
         =`omega[ sum_(i=1)^n m_i r_i^2]`
`rArr` L = `omega` I
where I = `sum_(i=1)^n m_ir_i^2` = moment of inertia of the rotating rigid body about the axis of rotation.

Problem to find the angular momentum of a cylinder:
A solid cylinder of mass 200kg rotates about its axis with angular speed 100 s^-1  .The radius of the cylinder is 0.25m.What is the magnitude of the angular momentum of the cylinder about its axis?
Given data : Mass M = 200 kgs
Angular speed `omega` =100 s^-1
Radius R=0.25m
L= ?
Formulas : I = `(MR^2)/2`
L = I`omega`
Working: I = `(MR^2)/2= (200 xx (0.25)^2)/2 = 6.25 "kg" m^2`
L = I`omega` = 6.25 x 100 = 625 Kg m² s^-1

Check my best blog Atomic Number of Curium.

Atomic Number of Curium

Curium is a silvery metal that is hard and brittle and tarnishes gradually in air at room temperature.

Introduction to atomic number of curium

It is first produced by Glenn T. Seaborg, Albert Ghiorso, and Ralph A. James at University of California in 1944. Its symbol is Cm and the atomic number of curium is 96. Curium does not occur in nature and is a synthetic chemical (produced artificially) produced in nuclear reactors by bombarding plutonium with helium ions (alpha particles).

Properties of Curium

• Molecular Weight: 247.070347 [g/mol]
• IUPAC Name: curium
• Canonical SMILES: [Cm]
• InChI: InChI=1S/Cm
• InChIKey: NIWWFAAXEMMFMS-UHFFFAOYSA-N
• Atomic number of curium: 96
• Element category: actinide
• Period and block: 7, f
• Electron configuration: [Rn]7s25f76d1
• Phase: solid
• Density: 13.51 g•cm−3
• Melting point: 1613 K
• Boiling point: 3383 K
• Crystal structure: hexagonal close-packed
• Atomic Radius: 170 pm
• Oxidation States: 3

Uses of Curium

• Curium is available only in extremely small quantities. Curium can be used as source of thermoelectric power in crewless space probes and satellites without any heavy shielding.
• Curium-242 isotope is used in radio isotopic power generators as it produces around 3 watts of heat energy per gram (through radioactive decay).
• Curium-242 is used as source of alpha particles in lunar missions to bombard alpha particles to the moon’s soil to determine materials present in moon soil.

Isotopes of Curium

About sixteen different isotopes of curium are present and some of the main isotopes are Cm-242, with half life of 160 days, Cm-243, with half life of 29 yr, Cm-244, with half life of 18 yr, Cm-245, with half life of 8,500 yr, Cm-246, with half life of 4,700 yr, Cm-247, with half life of 16 million yr, Pu-243, with half life of 5.0 hr, Cm-248, with half life of 340,000 yr, Cm-250, with half life of 6,900 yr, Pu-246, with half life of 11 days, Bk-250, with half life of 3.2 hr, and Am-246, with half life of 39 min.

The Chemical Formula for Aluminum

Aluminum belongs to group 13 of the periodic table. It was first discovered by Wohler in 1827. It is the third most abundant element in the earths crust. The atomic number of aluminum is 13. Aluminum forms a tri-positive ion i.e. Al3+. It is less electropositive than sodium and magnesium.

PROPERTIES OF ALUMINIUM

PHYSICAL PROPERTIES

1) Aluminum is a soft silvery white metal. The fresh metal on exposure to moist air loses its shining due to formation of oxide layer on its surface.

2) It is very light metal with specific gravity equal to 2.70.
Chemical Properties of Aluminium

(1) Action of air.

(a) Aluminum is not affected by dry air but in moist air a thin film of oxide is formed on its surface.

(b) It burns with oxygen with a brilliant white light with the evolution of heat

4Al +3O2 ----> 2Al2O3

(2) Action of water: Aluminum is not affected by pure cold water. However, saline water corrodes rapidly especially when it is hot. It decomposes boiling water. In the form of amalgam, it reacts more easily and rapidly with water and can decompose it even in cold.

2Al + H2O ----> 2Al (OH) 3+ 3H2

(3)Action of acids: Aluminum dissolves in dilute hydrochloric acid forming aluminum chloride with the evolution of di-hydrogen gas.

2Al + 6HCl  ----> 2AlCl3 + 3H2

Aluminum is not attacked easily by dilute sulphuric acid. This probably is due to insolubility of oxide layer (present on its surface) in the acid. However, it dissolves in hot and concentrated sulphuric acid to form sulphur dioxide.

2Al + 6H2SO4 ----> Al2 (SO)3 + 2SO2 + 6H2O
Uses of Aluminium

1)     Auminium is used for making electrical transmission cables.

2)     Aluminium powder is used as a reducing agent inj Goldschmidt aluminothermic process and thermite welding.

3)     It is used in making household utensils and novelty articles.

4)    Aluminium foil is used for wrapping soaps, cigarettes, confectionary, etc.

5)     It is used for making silvery paints for covering iron and other materials.

6)     It is used as dexidiser for removing blow holes in metallurgy.

7)     Due to low density, good thermal and electrical conductivity and resistance to corrosion, aluminium is used for making several alloys which are extensively used in automobile and automobile industries.

Chemical Reactions

The process of transformation of chemical substance to another is known as chemical reaction.

A chemical change involves the formation and cleavage of chemical bonds between atoms. Any chemical change can be described by using a chemical equation which gives complete idea about conversion of molecules during reaction.

Let’s elaborate; what is a Chemical reaction. It can define as change in bonding of reacting molecule to form new chemical compound by the formation of new chemical bonds.

The chemical substances take part in chemical reactions is called as reactant and newly formed substances are known as products. A chemical change can complete in one step or multi steps which can be described by using reaction mechanism. Chemical equation is the graphical representation of such a reaction which provides direction of reaction, physical states of reactant and products and the number of molecules taken part in reaction.

Different Types of Chemical Reactions and examples are as follow;

1. Combustion: Oxygen combines with substance to form carbon dioxide and water with a large amount of heat.

CH₄+ 2 O₂  CO₂ + 2H₂O

2. Synthesis: two or more chemical substance combines to form a more complicated one.

2Mg + O₂ 2MgO

3. Decomposition: A complex molecule decomposes in to simpler ones.

2HI  H₂ + I₂

4. Single displacement: one element trades places with another element in a compound.

Mg + 2 HClMgCl₂ + H₂

5. Double displacement: The anions and cations of two different molecules interchange and form two entirely different compounds.

BaCl₂ + H₂SO₄  BaSO₄ + 2HCl

6. Neutralization: A double displacement reaction between acid and base to form salt and water.

HCl + NaOH  NaCl + H₂O

There are a many observations which indicate a chemical change has occurred. These are called as Signs of a Chemical changes. Like; the formation of a precipitate during chemical reaction is one the best sign which can observe easily. When an ionic compound reacts with other to form insoluble salt, it gets settle down at the bottom of test tube in the form of precipitate. Other signs of chemical-reactions are;

• Color change.
• Liberation or absorption of energy.
• Formation of gas.
• Change in temperature of reacting solution

Let’s perform any Chemical Reaction Experiments and observe sign of reaction.

Take 1-2 ml of a 0.1 M lead (II) nitrate solution in a test tube and add it to 1-2 ml of a 0.1 M potassium iodide solution. When lead nitrate reacts with potassium iodide it form yellow colour precipitate of lead iodide it’s an example of double displacement reaction.

Pb(NO₃)₂ + 2KI  PbI₂ + 2KNO₃

Hence color change and precipitate from colorless reactants can be observed in reaction.

Moment of Inertia Sphere

The moment of inertia of a point mass about a known axis is defined by I = mr²  where m is its mass and r is its perpendicular distance from the axis of rotation.

Introduction :

Definition :  The moment of inertia of a rigid body about an axis is defined as the sum of the products of the masses of different particles, supposed to be constituting the body, and the square of their respective perpendicular distances from the axis of rotation.
                     Moment of inertia of a sphere can be explained in two parts (1) Solid Sphere (2)Hollow Sphere.
(1) Moment of inertia of a Solid Sphere  :  
    (a) About an axis passing through its diameter :   Consider a solid sphere of mass M and radius R. Its moment of inertia about an axis of rotation passing through its diameter is
                                                                            I  =  MR²
   (b) About an axis passing through its tangent  :   Let A'B' the tangent to the solid sphere. A parallel axis through its centre of mass is AB

By parallel axes theorem,
 Moment of inertia about the tangent = Moment of inertia about a diameter + Mr² .
                                                                  =`(2)/(5)` MR² + MR²
                                                         I      =  `(7)/(5)` MR² .

Moment of Inertia Sphere : Hollow Sphere

(2) Moment of Inertia of a Hollow Sphere
(a) Moment of inertia about an axis passing through the diameter of a hollow sphere of mass M and radius R is
                                                I  =`(2)/(3)`   MR² . 
(b)  Moment of inertia about an axis passing through its tangent can be obtained by applying parallel axes theorem. It is given by
                                                  I  =  `(5)/(3)` MR² .

Moment of Inertia Sphere : Example Problem

Problem : If the radius of the earth is suddenly halved keeping its mass constant, find its time period of rotation around its own axis.
Solution :  When the radius of the earth gets reduced suddenly keeping its mass constant, the angular momentum of the earth remains constant.      
                                                I   = constant
     If I changes from I₁ to I₂ ,   changes from`omega` ₁  to`omega`  ₂ so that 
                                          I₁ `omega`₁  =  I₂`omega`₂ .
     Assuming the earth to be a solid sphere, its moment of inertia about its diameter,
                      I  =  MR²
   If the radius changes form R to R/n
                                           `(2)/(5)` MR₁² `omega`₁  =  `(2)/(5)` MR₂² `omega`₂ 
                                                           R²  =  `(2pi)/(24 hours)`  = [R / n]² `(2pi)/(T)`  
          The time period of rotation , T = 24 hours / n² .
      In this problem, the radius changes from  R  to R / 2 .
                         `:.`   `(2)/(5)` MR²  `(2pi)/(24 hours)`   =  `(2)/(5)` M [R/2]² `(2pi)/(T)`
                               T  =  24 / 2²   =  24 / 4  =  6 hours.

Alkaline Metal facts

Introduction :

Alkaline metals are of very keen importance to us. These metals were discovered in the first decade of 19th century by an English chemist Sir Humphry Davy (1778- 829). Along the same time, he also found some elements of other metal families. Alkaline metal facts include all the basic properties shown by them.

About Alkaline Metals
Alkaline metals constituting Group2 of the Periodic Table. These metals show some general properties which are as follows:

• These metals are softer than most other metals
• These metals readily react with water (especially when heated).
• These metals are powerful reducing agents.
• These metals form divalent compounds, etc.

Facts Related to Alkaline Earth Metals

There are various facts related to Alkaline earth metals, some of the important ones are as follows:

• The name alkaline metals owes to their oxides that simply give basic alkaline solutions. These metals melt at high temperature and remain solids in heated atmospheres.
• The alkaline metals show good trend in their properties in the periodic table, with well-defined homologous behavior in going down the group.
• Some alkaline metals like Be and Mg, they show a distinguishable flame color , brick-red for Ca (Calcium) , Magenta-Red for Sr (Strontium) , Green for Ba (Barium) and crimson red for Ra (radium).
• The metals coming in this group show patterns in their electronic configuration, especially the behavior of them in their outermost shells, which results in the trend in chemical behavior.
• The alkaline metals are mostly Silver colored, soft metals, which react readily reactions with halogens.
• Alkaline Metal like Beryllium is highly toxic, it is rarely available to biological systems, and it has no known role in living organisms.
• Other Alkaline metal like Magnesium and calcium are essential to all known living things. They are involved in many roles like in some cellular processes, Mg functions as the active centre of the enzymes and Ca salts takes structural roles.
• Strontium and barium have lower availably in the atmosphere. They play very important roles in marine aquatic life, especially hard corals. These two are also used in medicines. Strontium is used to build the exoskeleton.
• The last metal that is Radium has a low availability and is very highly radioactive.

Conclusion

Alkaline earth metals are of very keen importance to us. These metals are highly important for the automobile industries due to their structure qualities. These metals are of great concern, as they are also included in building machines and other important equipment.