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Tuesday, 8 October 2019

Balancing of Several Masses Rotating in Different Planes

When several masses revolve in different planes, they can be transferred to a 'Reference plane' (R.P), which may be defined as the plane passing through a point on the axis of rotation and perpendicular to it. The effect of transferring a revolving mass (in one plane) to a reference plane is to cause a force of magnitude equals to the centrifugal force of the revolving mass to act in the reference plane, together with a couple of magnitude equal to the product of the force and the distance between the plane of rotation and the reference plane. In order to have a complete balance of the several revolving masses in different planes, the following conditions must be satisfied:
1. The resultant force must be zero (i.e All the forces in the reference plane must be balanced)
2. The resultant couple must be zero (i.e The couple about the reference plane must be balanced)
Let us now consider four masses revolving in different planes 1, 2, 3 and 4 respectively as shown in Fig., (a), and their relative angular positions are shown in Fig (b).,

          The magnitude of the balancing masses mand min planes A and B may be obtained as discussed below.
1. Take one of the plane say A, as reference plane (R.P).
2. The distances of the other planes to the left of the reference plane is taken as negative, and those are present on the right as Positive.
3. Tabulate the planes data in the same order from left to right as shown in below table.

4. A couple may be represented by a vector drawn perpendicular to the plane of couple. Couple C1 is obtained by transferring mto the reference plane through O. The couple obtained is m1 . r1 . land it acts in a plane through and perpendicular to the paper. The vector representing this couple is drawn in the plane of the paper and perpendicular to Omas OCshown by in Fig., Similarly the remaining vectors for remaining masses is calculated and shown in Fig.,
5. The couple vectors as discussed above, are turned counter clockwise through a right angle for convenience of drawing without changing relative positions.
6. Now draw the couple polygon as shown in Fig., The couples about the reference plane must be balance. i.e., the resultant couple must be zero.

7. Now draw the force polygon as shown in Fig., The forces in the reference plane must balance. i.e., the resultant forces must be zero.

From the above expression, the value of balancing mass min the plane 'A' may be obtained and the angle of inclination of this mass with the horizontal may be measured from Fig., (Angular positions of masses


Wednesday, 11 September 2019

HEIGHTS AND DISTANCES


          The topic of height and distance in trigonometry is an important topic in competitive examinations point of view. Generally we have seen the problems where the height of a building is given and then from the top of this building the angles of elevation or depression are given for another building and we have to find the height of the second building. In this article we will cover these type of problems.

There are certain terms associated with the heights and distances which are described as follows:

Angle of Elevation: Let us consider a situation where a person is standing on the ground at point ‘O’ and he is looking at an object which is at some height (above the level of his eye) say the top of the building (P). The line joining the eye of the person with the top of the building is called the line of sight. The angle made by the line of sight with the horizontal line is called Angle of Elevation.
In this figure the line of sight is making an angle θ with the horizontal line. This angle is the angle of elevation.
Angle of elevation of P from O = AOP

Angle of Depression: Now let us take another situation where the person is standing at some height (O) with respect to the object he is seeing (P). In this case the line joining the line of sight of the man with the object is called the line of sight. The angle made by the line of sight with the horizontal line is called angle of depression.
 In the above figure ‘θ’ is the angle of depression.

Note: The angle of elevation is equal to the angle of depression.

The questions on this topic require some basic knowledge of Trigonometry. We should be aware of the basic trigonometric ratios and their values.
          Let us recall that the ratios of the sides of a right angled triangle are called trigonometric ratios. These are sine, cosine, tangent, cosecant, secant and cotangent.
Let the ΔABC is a right angled triangle.

Then Sin θ = Opp/Hyp = AB/CB
Cos θ = Base/Hyp = AB/CB
Tan θ = Opp/ Base = AB/AC
Cosec θ = CB/AB
Sec θ = AB/BC
Cot θ = AC/AB

Also we should know the values of these trigonometric ratios of some common angles as given in the following table:

0°
30°
45°
60°
90°
Sin
0
1/2
1/√2
√3/2
1
Cos
1
√3/2
1/√2
1/2
0
Tan
0
1/√3
1
√3
Infinite
The values in the given table will be useful while solving the questions on height and distances.
After going through some examples we will learn how to measure the height and how to find the distance.

Friday, 25 January 2019

Electrification of a Body (or) Electrification processes of bodies

          A body can be charged or electrified in many ways; the methods are discussed below:
a. Charging by friction
b. Charging by conduction or contact
c. Charging by induction
d. Electrification byh heating
e. Electrification by pressure

a. Charging by friction: When two nutral bodies are rubbed against each other, due to friction one of them losses electrons and hence gets positively charged and the body that ains electrons gets negetively charged.
Experiment: Take a small piece of uncharged plastic rod and cloth, rub against each other this will result both of them get charged. Same thing will happen between rubber rod and fur, Glass rod and silk cloth.

b. Charging by conduction: When a charged body is brought in direct contact with an uncharged body (or nutral body), it shares its charge with it. Thus the uncharged body becomes charged.

c. Charging by Induction: The process of temporary electrification of an initially neutral conductiong body (induced) by bringing a charged body (inducer) close to it without making any actual contact between the bodies is known as charging by induction.

d. Electrification by heating: Certain bodies, when heated, is electrified by presenting contrary electricities names in two diametrically opposite points. The phenomenon is called pyroelectric phenomenon. It is more common in crystalls, such as Tourmaline.

e. Electrification by pressure: Certain bodies, when compressed, is electrified, displaying electricities names against the ends. The phenomenon is called piezoelecttric phenomenon. It is also ore common in crystals, such as Tourmaline, Calcite, and Quartz. 

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