**Unit and dimension notes**

**Examples of nuclear force in daily life**

(a) The nuclear force is responsible for keeping protons, neutrons etc. inside the nucleus.

(b) The phenomenon of radioactivity is the result of inadequate nuclear force between protons and neutrons inside heavy nuclei.** Weak Nuclear Force:** Another kind of force is encountered when reactions involving protons, electrons and neutrons take place. The weak nuclear force appears only in certain nuclear processes such as the β decay of the nucleus. These are very short-range forces much smaller than the size of protons or neutrons. The effect is experienced inside these particles.

They are 10^{25} times stronger than gravitational forces and weaker than electromagnetic and nuclear forces.**Laws of Conservation****(i) Law of conservation of linear momentum:** If no external force acts on a system the linear momentum of the system remains constant.

**Examples from daily life:**

(a) When a heavy nucleus disintegrates into two smaller nuclei the products move in opposite directions.

(b) When a bullet is fired from the gun, the recoil for the gun can be explained on the basis of this law.

(c) Two billiard balls, after the collision, obey this law.

(d) The motion of the rocket is obeyed by this law.

(ii) Law of conservation of energy: Energy can neither be created nor destroyed, however, it may change from one form to another.**Examples from daily life:**

(a) When a body falls freely under gravity, its mechanical energy (potential + kinetic) remains constant throughout.

(b) The mechanical energy of vibrating pendulum remains the same.

(iii) **Law of conservation of angular momentum:** If no external torque acts on a system then the total angular momentum of the system always remains constant.**Examples from daily life:**

(a) When a planet moves around the sun in an elliptical orbit, its velocity increases when it is close to the sun and decreases when it is far away from the sun.

(b) A diver jumping from a springboard exhibits somersaults in the air before touching the water’s surface.

(c) A ballet dancer increases her angular velocity by folding her arms and bringing the stretched leg close to the other leg.

(d) The inner layers of whirl-wind in a tornado have extremely high speed.

**Physical Quantity**

(Quantities in terms of which laws of physics are described and whose measurement is necessary)

**Unit:-** Unit of a physical quantity is defined as the reference standard used to measure it. The magnitude of a physical quantity is the product of a unit (*v*) in which quantity is measured & the number (*n*) of times that unit is contained in the given quantity.

**Practical Units**

For the measurement of a very large distance, the following three units are used:—

1. Astronomical Unit (AU): It is the average distance of the centre of the sun from the centre of the earth.

AU = 1.5 × 10^{11} m^{2}. Light Year: It is defined as the distance travelled by light in a vacuum in one yearlight year = 3 ×108 (365 × 24 × 60 × 60) merely = 9.4 ×10^{15} m(^{3})

1 Par sec = 3.1 × 1016 m

**Dimensional ****Analysis**

**Dimensions :**

Dimensions of a physical quantity are defined as the power to which the fundamental units of mass, length, time etc. must be raised to represent a derived unit of the quantity

So speed is said to possess zero dimension in mass, one dimension in length and –1 dimension in time.

Dimensional formula :

It is an expression which shows how and which of the fundamental units are required to represent the unit of a physical quantity.

is the dimensional formula of speed.

**Dimensional Equation : **

It is defined as an equation obtained by equating the physical quantity with its dimensional formula.

The dimensional equations of speed are

Based on dimensional analysis, physical quantities can be divided into four types—

1. Dimensional constants: Physical quantities which possess dimensions and have a constant value.

e.g., (i) Gravitational constant

(ii) Planck’s constant

2. Dimensional Variable: There are physical quantities which possess dimensions and do not have a constant value.

e.g., velocity, impulse, density.

3. Dimensionless constant: There are physical quantities which do not possess dimension and have a constant value

e.g., (a) Numerical constant 1, 2, 3

(b) Mathematical constant.

4. Dimensionless Variable: There are physical quantities which do not have a constant value. e.g. angle, strain and specific gravity.**The principle of Homogeneity of Dimensions:**

A given physical relationship is dimensionally correct if the dimensions of the various terms on either side of the relationship are the same.

e.g., V = u + at

The dimensions of every term in the given physical relation are the same. So according to the principle of homogeneity of dimensions, the relation v = u + at is dimensionally correct.

**Use of Dimensional Equations**

To convert a physical quantity from one system of units into another.

To establish a relationship between different physical quantities.

**Limitations of Dimensional Analysis**

Though the dimensional method is a simple and a very convenient way of finding the dependence of a physical quantity on other quantities of a given system, it has its own limitations, some of which are listed as follows:

♦In more complicated situations, it is often not easy to find out the factors on which a physical quantity will depend. In such cases, one has to make a guess which may or may not work.

♦This method gave no information about the dimensionless constant which has to be determined either by experiment or by a complete mathematical derivation.

♦This method is used only if a physical quantity varies as the product of other physical quantities. It fails if a physical quantity depends on the sum or difference of two quantities.

can not be derived by using the method of dimensions.

♦This method will not work if the quantity depends or another quantity is sine or cosine of an angle, i.e. if the dependence is a trigonometric function. The method works only if the dependence is by power functions only.

♦ This method does not give a piece of complete information in cases where a physical quantity depends on more than three quantities because by equating the powers of M, L and T, we can obtain only three equations for the exponents. For example, if for a simple pendulum, we assume that the time period T depends on angle* θ* of the swing, besides the factors I, m and g; then

By equating the dimensions on both sides, we would find that a, b and c can be determined, but d remains undetermined. Besides the quantities used in Mechanics, the other fundamental quantities like current, temperature and substance are dimensionally represented as A, K or and mol respectively. Thus dimensions of electric and thermodynamic quantities can also be determined. For example: