Concept of Friction

Friction and its Significance ::

Friction can be understood very easily with the help of an example...

Case (1) :: Suppose we are going in a desert area on foot or in a vehicle

Case (2) :: Suppose you are going on a road where small stones are crushed and laid down on the road

Case (3):: Suppose you are going on a plain road

Case (4) :: Suppose you are walking on tiles which are polished

Can you notice the difference in four cases then the topic of friction is very easy if not notice whatever you can experience in your day to day life

The repulsion ( force) we experience decreases from case (1) - case (4)

The Concept of friction can be understood well when we relate it to real life situations.

Friction :: It is the force which opposes the motion of an object or a person when force is applied.
   
It arises due to the surfaces in contact and the decreasing surface area of contact decreases the friction.

The same you can observe when a block of same area is cut in the shape of a rectangle and a circle, if you try to drag them along the edges i.e.,along the diagonal to the surface then the circular shape will move easily compared to the rectangular one.

The friction is the force which is responsible for the way we are walking,running or any other activity we are doing with our legs.

The effect of friction can be reduced to a large extent by applying lubricants...
Suppose you are walking on tile floor and pour oil in it and try to put your leg on that..
Never do this because you will end up with injuries because the frictional force is reduced which causes you to drag along the surface which is not a good thing.

Transfer of Motion

What are the things which drives a motion and what is the governing force behind the motion?

Solution:: The answer begins with a toy car which we played with in our childhood. How does the car moves?? First we compress the spring of the car and then when we release the turning movement then the car starts moving in forward direction because the spring which is compressed will come to its normal position. Here we are, again the spring is coming to its normal position then how come the wheels move??... The wheels are moving because of the gear box which is present inside the car...The gear box allows the car wheels which are connected by the shaft will move forward with the rotation of the shaft.

For any object when does the motion start?? The answer is quite simple when an external agent applies some force or some deviation to the system ,then the object will be set into motion with the help of many connecting elements...

What are these connecting elements??

Generally a shaft which connects the wheels or the motor fan which rises/lowers the fluids from a tank and so ...

What is the principle applied in operating a device??

The answer is quite simple...law of conservation of energy is the one which drives this mechanisms. Energy in one form is always getting converted to other and can be utilized with the help of suitable mechanisms which can make the system work. The driving elements should be identified so that the system will work for a long run.

The forces which we come across in designing large vehicles are the air drag which continuously acts on the system and the frictional forces which are present oppose the motion to a large extent as if the frictional force is always inversely related to the size of the machine and even the drag to the same extent.

Rain Drops are Spherical

Why does the rain drops are spherical??

Answer:: The property which is responsible for this is surface tension.

Surface Tension is defined as the force acting over a unit length

For a given volume only sphere has the minimum surface area compared with other geometries.

The cohesive forces which are present between the molecules will tend to occupy a minimum surface area because of which there exists a pressure difference between the inner and outer molecules of the surface which can be calculated as ∆P=4T/r, T is the surface tension and r is the radius of the drop.

The drops generally of 1mm radii are spherical in shape and bigger drops tend to flatten on the gottom and curved on the top, this is due to air drag due to friction

The size of rain drops generally vary from 0.02 inch to 0.33 inch in thunderstorms.

Distance between two points when viewed in Google earth

Question:: Calculate the distance between two points when viewed through Google earth

Solution:: First we have to know how to locate a point in google earth,
                  We locate any point with the help of the latitudes and longitudes

The latitudes and longitudes are specified using the degrees measurement

The two points are indicated by (lat1,long1), (lat2,long2)

The formula is given by haversine.

Distance between two points

Question:: Calculate the distance between two points normally and on google earth.

Solution :: The concept of distance between two points in a coordinate plane can be understood using a coordinate plane in 2-D and 3-D

In a 2-D plane the distance between two points can be calculated as
General point in a 2-D plane can be expressed as (x,y)
Take the two points to be calculated as (x1,y1) and (x2,y2)
The distance between two points can be calculated as

                                 Distance between the points A and B can be calculated by using the right
                                  angle triangle ABC
                                 Distance between two points is given by
                          

In a 3-D plane the distance between two points can be calculated as
General point in a 3-D plane is expressed as (x,y,z) in rectangular coordinate system

Construct a cuboid using the given points A and B

Distance between the two points is given by

Motion of tyres around a Circular Path

Question :: What is the motion of tyres around a circular path and explain about their velocities?

Solution :: The motion of different tyres can be understood by taking different examples
1.) 2 - wheeler
2.) 3 - wheeler
3.) 4 - wheeler

1.) 2 - wheeler
      When a 2- wheeler is going around a circular path then the tyres will be moving in a circular path with equal radii, but the weight of the body creates a net torque when moving in a circular path.

2.) 3 - wheeler
       When a 3 - wheeler is going round in a  circular path then the tyres tend to rotate in different radii, the rear wheel which is close to the turning will move in a smaller radius when compared to the front wheel, another wheel on the rear side. The three wheels rotate in three different radii and the 3 tyres move with different velocities. The three velocities can be varied by a shaft and gears connecting  them.

3.) 4 - wheeler
    This is similar three wheeler on the rear side but the front side is behaving same as the rear side, the wheels near to the circular arc tend to rotate with smaller radius there by smaller velocities compared to the wheels which are away from the circular turn. The speeds of the tyres can be varied by a differential gear attached to the shaft.

Angle in a Regular Polygon

Finding the sum of the angles in a regular polygon

Solution:: The polygon refers to a figure which is closed and regular polygon means all the sides are of equal length.

3 - equal sides    ::   Equilateral triangle
4 - equal sides    ::   Square
5 - equal sides.   ::   Regular Pentagon
6 - equal sides    ::   Regular Hexagon

Sum of the angles in a triangle is 180°
                                 in a square is 360°
                                 in a pentagon is 540°
                                 in a hexagon is 720°
Sum of the angles in a n - sided polygon is (n - 2)*180°


In a regular polygon all the sides are equal which tells us all the angles are equal.

Obtaining an angle for a regular polygon :: First we will start with a triangle and we will
proceed further with the other regular polygons.

The sum of the angles in a triangle is 180°
The proof is given in SUM OF THE INTERIOR ANGLES OF A POLYGON

For a 4 - sided polygon number of triangles are 2,
          5 - sided polygon number of triangles possible are 3,
          6 - sided polygon number of triangles possible are 4

 As we sum it up for a n - sided polygon number of triangles are (n -2), n greater than or equal to 3.

Sum of the interior angles of the n - sided polygon are (n -2)*180°

Each angle in a regular polygon of n- sides is (n -2)*180°/n, n is greater than or equal to 3

Sum of the interior angles of the Polygon

Find the sum of the angles of the polygon

Solution :: First we find the Sum of the interior angles of the triangle and we relate it further to the polygons with 4,5,6 sides and so on...

For a quadrilateral : Number of triangles possible are 2,
For a pentagon.     :  Number of triangles possible are 3,
For a hexagon.       : Number of triangles possible are 4,

For a n- sided polygon number of triangles possible are (n-2)

The sum of the interior angles of a n-sided polygon are (n-2)*180° = (n-2)*π radians.

Areas Applications

Find the area of the figures from the geometry you know so far.

Solution :: The given figures are 1. Regular Hexagon, 2. Regular Pentagon 3. Area of the shaded part   4. Area of the shaded part

Area of Triangle applications

Some figures are given. Find the areas of them using the geometry known to you so far.
Using area of triangle

Solution :: The area of the figures can be found

Pascal Triangle

Do you know what is a Pascal triangle and what is its use?

Solution :: A Pascal triangle is used to calculate the binomial coefficients without much
                   struggle. The triangle is drawn as follows.
                   Start with 1 (This is the first row element,which can be said as 0! )
                    Now the second row can be obtained by adding the two elements of the first row
                     (i.e., 1 and 1)
                    The third row is obtained by writing the end terms as 1 and the remaining by
                     adding the terms in the previous row and which are present above it.
                     The subsequent rows can be obtained in the same manner...
                                                                    1
                                                                1 -    1
                                                           1 -    2 -    1
                                                       1 -  3 -     3 -    1
                                                         (1+2)   (2+1)
                                                   1 -   4 -     6 -     4 -   1
                                                     (1+3)  (3+3) ( 3+1)
                                              1 -    5 -     10 -    10 -   5 -   1
                                                   (1+4) (4+6) (6+4) (4+1)
                                        1  -   6  -   15  -    20  -    15   -    6 -   1
                               
                                   1 -    7 -   21 -    35 -   35 -     21 -    7 -   1

                               1 -     8 -   28 -   56 -   70 -    56 -    28 -   8 -     1

Length of Altitude in a right angled triangle

Find the length of the altitude of a right angled triangle where the sides are given by 'a' and 'b'.

Solution:: There are many ways to solve this problem....
Method :1 Similar Triangles Concept

Method 2 :: Trigonometric Approach

Method 3 :: Pythagoras Theorem

Method 4 :: Straight Lines Approach

Method 5 ::  Area of Triangle Approach

Division of a Circle

Divide a circle into 'n' parts and what is the geometry of each part?

Solution:: Suppose if a line is drawn through the centre of a circle then it will divide it into 2 equal parts and if you draw 2 lines at right angles passing through the centre then it will divide into 4 equal parts...and for 3 lines at 60° each then 6 parts and it goes on...

For odd number of parts what we have to do??

It's simple as it sounds because initially we used angle concept.. The same can be applied here also.. If we want 3 equal parts then the angle at the centre should be divided by 3 , for 5 parts the angle at the centre is divided by 5 and so on..

If you want to divide into 'n' parts then the angle at the centre is divided by 'n'.
The angle of each part is (360/n)° and all the parts are sectors with equal area.

Let's see this diagrammatically...

Concept of Relative Motion

What is relative velocity and how can we understand it in a easy manner.

Relative Motion of two bodies :: Suppose if two bodies are in motion and are separated by a  distance then the concept of relative motion is useful.

Relative Velocity:: Suppose A and B are two objects which are moving with velocities 

X m/sec and Y m/sec then we can apply relative velocity of the two bodies . we can analyse the motion of the bodies.

Even we have to look for the motion of the objects A and B whether they are travelling in a straight path (1-d)or not(2-d / 3-d).

In 1 - D plane the explanation can be understood like this...

Suppose if A and B are approaching each other then the distance between them decreases as because their relative velocity is more compared to their original velocities.

Suppose if A and B are travelling in the same direction then the distance between them increases or decreases depending upon the velocities X and Y.

The whole situation can be viewed as