Units and Dimensions, Physics tutorial

Units: 

A unit has to be stated prior to any type of measurement can be made. You are aware that various systems of units have been employed in the past. Though, the system which consists of gained universal acceptance is the System International d' Unites generally termed as S. I. Units (adopted in the year 1960). The S.I Unit is mainly based on the meter as the unit of length; the kilogram (kg) as the Unit of Mass; the second (s) as the Unit of Time, the ampere (A) unit of current and the Kelvin (K) unit of temperature.

Dimensions:

We frequently employ the word dimension in Physics to explain the relationship between a physical quantity and the fundamental quantities deduced in terms of the symbols M, L, T of the fundamental quantities mass, length and time correspondingly. You must note that physical quantities can either be dimensional or dimensionless.

Dimensionless Quantities don't base on the system of unit in which they are measured. That simply signifies that they encompass no units. Illustrations are angles and their trigonometric ratios (that is, ratio of two lengths); relative density (or specific gravity), efficiency of a machine (that is, ratio of two quantities of work).

Dimensional Quantities on the other hand based on the magnitude of the fundamental units in which they are calculated and are dissimilar in various unit systems. Let consider that the fundamental quantities are symbolized by the symbols M, L, and T, you will be capable to find out (or derive) the dimensions of Physical quantities.

For example: Area (length x breadth) has dimension of L x L = L2

Fundamental Quantities and their Units:

  • Fundamental quantities are the essential quantities on which other quantities based.
  • Fundamental units are the essential units on which the other units based on. These are the units of the fundamental quantities.

Quantity

Unit

Unit abbreviation

Length

Metre

m

Time

Second

s

Mass

Kilogram

kg

Electric Current

Current

A

Temperature

Kelvin

K

Luminous Intensity

Candela

cd

Amount of substance

Mole

mol

Derived Quantities and their Units:

The units of all physical quantities that are mainly based on the three fundamental units are known as derived units. This is how to get derived unit from the fundamental unit. The unit of area is the area of a square each side of which is of one unit length.

Derived Quantity

Derivation

Derived Unit

Area (A)

Length x Breadth

m2

Volume (V)

Length x Breadth x height

m3

Density

Mass/Volume

kgm-3

Velocity (V)

Displacement/time

ms-1

Acceleration (a)

change in velocity / time

MS-2

Force (F)

Mass x acceleration

Newton (N)

Energy or Work (W)

Force x distance

Joule, J (Nm)

Power (P)

Work/time

Watt, W (J S-1)

Momentum

Mass x velocity

Kg. m. s-1, Ns

Pressure (P)

Force/area

N m-2 (Pascal)

Frequency (f)

number of oscillation/ time

Per second or s-1 (Hertz, HZ)

Electric charge

∫ idt

Coulomb (c)

Electric potential difference

Work/charge

Volt (V)

Elector motive force

Work/charge

Volt (V)

Electric resistance

Electric-potential difference /current

Ohm (?)

Electric capacitance

Charge/Volt

Farad (F)

Scalars and Vectors:

Concept of Scalars:

The Scalars quantities are such quantities which encompass only magnitude or numerical value however no direction, illustrations are length or distance, volume, mass, density, work, time, speed, temperature and energy.

Concept of Vectors:

However, we are familiar that most of the measurable physical, quantities have magnitude or numerical value and also direction; these quantities are not fully explained unless their magnitudes and directions are specified. Such groups of quantities are recognized as vector quantities.

Illustrations are: Weight, displacement, velocity, force, acceleration, momentum, electric field and so on.

Vector quantities are such quantities that have both magnitude (size) and direction.

Difference between Scalars and Vectors quantities:

Scalar Quantity

Vector Quantity

1) A physical quantity which consists of only magnitude is termed as scalar quantity.

A physical quantity which consists of magnitude and also direction is termed as vector quantity.

2) For illustration: mass, length, time, work and so on.

For illustration: velocity, acceleration and so on.

3) Scalars follow arithmetical rules for the operations such as addition and multiplication.

Vectors follow vector algebra for the operations such as addition and multiplication.

4) A scalar can divide the other scalar.

Two vectors can never split or divide each other.

Addition and Subtraction of Scalars:

The process of adding and subtraction of scalars is straight forward. They add or subtract just similar to ordinary numbers by ordinary arithmetical methods.

Illustrations:

i) 5 cm3 + 7 cm3 = 12 cm3

ii) 5 kg - 2 kg = 3 kg

Addition and Subtraction of Vectors:

Dissimilar to the scalars, in adding or subtracting vectors, you have to think about both sizes and directions of the vector quantities beneath consideration.

Resultant vector: A single vector that consists of the similar effect as two or more vectors acting in similar direction is termed as the resultant vector. Addition of vectors gives mount to a resultant vector.

Vectors acting in the same line:

Two or more than two vectors acting in the similar direction might be added as if they were scalars. For illustration the sum or resultant of the three forces shown in figure below is 50 N acting right to left while in second figure it is 250 N left to right.

722_vectors in same line.jpg

Vectors Inclined at an Angle:

i) Using Pythagoras theorem:

The Pythagorean Theorem is a helpful process for finding out the outcome of adding two (and just two) vectors which makes a right angle to one other. The technique is not applicable for adding more than two vectors or for adding vectors which are not at 90o to one other. The Pythagorean Theorem is a mathematical equation which associates the length of the sides of a right triangle to the length of the hypotenuse of right triangle.

Illustration: Tom leaves the base camp and hikes 11 km, north and then hikes 11 km east. Find out the Tom's resultant displacement.

591_vectors-right triangle.jpg

ii) By using Parallelogram Law:

The method of 'the parallelogram of vectors addition method" is evaluated by the following steps:

  • Draw vector 1 by using suitable scale and in the direction of its action.
  • From the tail of vector 1 sketch vector 2 by using the similar scale in the direction of its action.
  • Complete the parallelogram by employing vector 1 and 2 as sides of the parallelogram.
  • The resultant vector is symbolized in both magnitude and direction by the diagonal of the parallelogram.

1901_Vectors-parallelogram.jpg

iii) By using Triangle law:

  • The method of 'the triangle of vectors addition method' is as follows:
  • Sketch vector 1 by using suitable scale and in the direction of its action.
  • From the nose of the vector draw vector 2 by using the similar scale and in the direction of its action
  • The resulting vector is symbolized in both magnitude and direction through the vector drawn from the tail of vector 1 to the nose of vector 2

2344_Vectors-triangle law.jpg

Resolution of Vectors:

The method of splitting or dividing a vector into different parts or components is termed as 'Resolution of Vector'. Such parts of a vector might act in different directions and are termed as 'components of vector'.

The parallelogram process of vector resolution comprises using a precisely drawn, scaled vector diagram to find out the components of the vector. In brief put, the method comprises drawing the vector to scale in the indicated direction, drawing a parallelogram around the vector in such a way that the vector is the diagonal of the parallelogram, and finding out the magnitude of the components (that is, the sides of the parallelogram) by using the scale. A step-by-step method for employing the parallelogram method of vector resolution is as follows:

i) Choose a scale and precisely draw the vector to scale in the pointed direction.

ii) Draw a parallelogram around the vector: starting at the tail of the vector, draw vertical and horizontal lines; then draw horizontal and vertical lines at the head of the vector; the sketched lines will meet up to form the parallelogram.

iii) Sketch the components of the vector; the components are the sides of parallelogram; be sure to place arrowheads on such components to point out their direction (up, down, left and right).

iv) Significantly label the components of the vectors by using symbols to point out which component is being symbolized by which side; a northward force component would be labeled Fnorth; a rightward velocity component may be labeled vx; and so on.

v) Measure the length of sides of the parallelogram and utilize the scale to find out the magnitude of the components in real units; label the magnitude on the diagram.

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