JEE advanced preparation tips for Unit & Dimension

JEE advanced preparation

In this post you will read about JEE advanced preparation tips for Unit & Dimensions. So let’s start.

JEE advanced preparation for Dimensions

Indirect and direct measures

In it’s simpler form, a measurement is the comparison of an experimental result with a pattern (unit of measurement). This means that when we say that something is 3 meters long what we really mean is that the measured length is three times the length of the pattern, which is 1 meter.

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From a series of direct experimental measures we can obtain indirect or derived quantities. For example, to measure the area of the floor of a rectangular room we only need to measure the length of two different sides and apply the formula S = bh. The existence of these relationships allows to define these magnitudes in fundamental and derived.

Dimensions of a magnitude

Independently of the unit that is used to express a physical magnitude, this are classified in different types, according to the way you can add them together. For example, we can add a distance of 3 km with one of 2 miles, or we can 5 kg to 3 pounds, but we know that it is wrong to try to add 5 kg to 3 km. We see that the is something more basic that the unit of measurement and that is the magnitude type: distance, weight, mass, time,… Each one of this types is called dimension and we say that a magnitude has “distance dimensions” or “mass dimensions”.

Dimensional homogeneity

To classify the magnitudes we have the dimensional homogeneity principle that establishes that:

In every equation and every sum, the equalized or added terms must have the same dimensions.

This is an elegant way of saying “you can not add pears with screws”. This principle constitutes an extremely useful tool to detect calculation errors. Let us imagine that, as a result of a problem, we arrive at the conclusion that the force is equal to:

F = A(r-r^{2})

being r a radius an A a constant. This equation is incorrect, without the need to substitute for any numerical value. We are adding a distance, r, that has length dimensions to a squared distance, r2, which is an area. Since these quantities have different dimensions, the equation is not valid.

Dimensional homogeneity allows to quickly locate mistakes in the results of an exercise.

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A relationship between magnitudes does not imply any concrete unit, only it’s dimensions. By saying that the distance between Daman and Kolkata is the same as Kolkata and Mumbai, it does not matter if the distance is measured in inches or kilometers. That is why it is not correct to write a law as:

E = mv2/2 (jules) → incorrect expression

since energy can be expressed as ergs, calories, kilowat.hour, and so on, depending on in which unit we measure the mass or the velocity. That is why the rule is that, if a formula is purely algebraic, you must not include the units. On the contrary, if you substitute one or every constant by their numeric value you must include their units.

Dimensional equations

Even thought different magnitudes can not be added, they can be multiplied. We can divide a magnitude with distance dimensions by one with time dimensions and we obtain as a result a magnitude with velocity dimensions. Let us write this relationship:

[v] = [x] / [t]

where the [] represents “dimensions”. We must insist that this equations does not tell us that the velocity is equal to the space divided by time, but that it’s units are those of a distance divided by the units of time (that can be m/s or km/h, for example).

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Dimensional homogeneity allows us to determine the dimensions of unknown quantities. For example, in Hooke’s law:

F = k.x

says that the k constant has dimensions of force times the distance.

[k] =[F]/[x]

that is, for example, N/m.

The existence of relationships between dimensions allows us to divide the magnitudes in two groups, fundamental and derived. From a relationship such as:

S = b.h

we understand that the dimensions of the area are those of a distance squared, which can be written as:

[b]=L , [h]=L,  [S] = L^{2}

In this way, the dimensions of any magnitude can be expressed as powers of a series of fundamental magnitudes.

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For example

the velocity equals the quotient of a distance divided by an time interval and, as such, the dimensional equation is verified:

[v] = \frac{[x]}{[t]} = LT^{-1}

Here, distance and time are considered fundamental magnitudes and the velocity is a derived magnitude.

The magnitudes that are chosen as fundamentals, and even their number, are arbitrary. In the IS we find seven fundamental magnitudes: length, time, mass, electric current intensity, amount of matter, thermodynamic temperature and luminous intensity. Everything else is derived.

Each derived magnitude has an only one dimensional equation, characterized by the different powers of the fundamental magnitudes.

Magnitude

Relationship

Dimensions

Area

[S] = [x]2

L2

Volume

[V] = [x]3

L3

Velocity

[v] = [x] / [t]

L.T-1

Acceleration

[a] = [v] / [t]

L.T-2

Force

[F] = [m][a]

M.L.T-2

Work

[W] = [F][x]

M.L2.T-2

Potency

[P] = [W] / [t]

M.L2.T-3

Having the dimensional equations of the different magnitudes that appear in an equation, we can establish in a systematic way if it is dimensionally correct.

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For instance, the equation for the velocity of a body that is going to impact with the ground is:

v = \sqrt{2gh}

being h the initial altitude, v the impact velocity and g the acceleration of gravity, it complies: 

\frac{L}{T} = \sqrt{\frac{L^{2}}{T^{2}}} = \frac{L}{T}

and as such it is dimensionally correct.

We must emphasize that the homogeneity is independent from the units employed to measure the quantities. As far as we know, h can be measure in inches, and v in microns per week. The dimensions of a magnitude are something more basic that the units we use to measure.

JEE advanced preparation : Measurement units

Measurement are arbitrary and, in most occasions, are defined with a concrete problem in mind. For example, when someone says that an accident happened midway between Kolkata and Daman, they are using the distance between this cities as a unit, and they are saying that the accident occurred in x= 0.5u.

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To be able to make results easily to interpret and be able to translate them to other situations, is better to employ a standardized unit system. Of the different systems of units in use, the most accepted is the International Unit System (IS), which evolved from the decimal metric system developed in France during the French Revolution.

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