QUANTITIES HAVING THE SAME DIMENSIONAL FORMULA

To define the dimensions of a physical quantity we have to raise the powers of the base or fundamental quantities, for example, length, time and mass, etc. are used to show the amount. Simply, the product of different powers is used to express a quantity in reference to base quantities. The base quantities are the prerequisites to derive the physical quantities.

One letter symbols are used to represent the base quantities. For example,

• Symbol M stands for mass.
• Symbol L stands for length and 
• Symbol T stands for time
• The symbol I stands for electric current
• Symbol K stands for thermodynamic temperature
• Symbol Mol stands for the amount of substance
• Symbol cd stands for luminous intensity

THE TERM- DIMENSIONAL FORMULA

The term ‘dimensional formula’ refers to the expression used for representing a physical quantity in reference to a base quantity. The word ‘quantity in that base’ refers to the exponent value of the base quantity. Following are some examples described where the quantities have the same unit and dimension. 

Check out the complete UPSC Syllabus

LINEAR MOMENTUM

The momentum (linear) of any particle is described as the product of the particle’s mass and the velocity by which the particle is moving.

The formula to represent linear momentum is 

(p= mv)

Where p is the symbol for linear momentum

m is the symbol for the mass of the particle

and v is the symbol for the velocity of the particle. When observing through an inertial frame, the particle has a constant momentum as time passes when the particle is following a uniform velocity. The Dimensional formula to express the quantity- linear momentum is described as [MLT-1].

IMPULSE

As per classical mechanics, impulse can be described as the impulse is the force (F), over the time (t) for which it acts. The SI unit of the physical quantity impulse is – newton second (Ns). The dimensional formula for physical quantity impulse is [M1 L 1 T-1]. 

IN CONTEXT TO THE DIMENSION OF FORCE

The word force is often used to indicate an action when we try to pull or push any object. By applying a push force, we tend to keep it away from us whereas when applying a pulling force, we tend to bring the object close to us. Simply we can absorb that force is a medium of interaction between the two objects. The SI unit for the quantity force is Newton (symbol ‘N’). The dimensional formula to describe the physical quantity force is M1 L1 T-2

Force can be classified into two large categories namely,

1. Contact forces: examples such as friction, the tension in a spring, spring force etc.
2. Non-contact forces: examples such as gravitational force, magnetic forces etc.

The four major types of forces that prevail in nature are as follows:

1. GRAVITATIONAL FORCE

• There is a presence of attraction between the two objects by the virtue of their masses.

2. ELECTROMAGNETIC (EM) FORCE

• Besides gravitational force, the two bodies might exert electromagnetic (EM) forces of some magnitude on one another because the bodies possess charge.

3. NUCLEAR FORCE

• When in this case, the interacting particles are protons or neutrons or both of them, and in other cases as well, nuclear forces come into action. Anything that involves changes within the nucleus, there is a presence of nuclear forces. 
• The phenomena of radioactivity, nuclear fission, nuclear fusion etc. result from nuclear forces.

4. WEAK FORCES

• The reactions that involve particles such as protons, electrons and neutrons usually involve weak forces.

Also see

• Tips for UPSC Preparation
• How to Prepare for UPSC without Coaching

IN CONTEXT TO THE DIMENSION OF ENERGY 

As per scientists, energy can be defined as the ability to perform work. Energy prevails in its various forms described as follows:

• Heat Energy
• Light Energy
• Motion Energy
• Electrical Energy
• Chemical Energy
• Gravitational Energy

The broad categories in which we classify energy for performing work is 

• POTENTIAL ENERGY: The energy possessed by a particle when it exhibits no motion or is at rest.
• KINETIC ENERGY: Quantitatively, it defines the energy of a moving particle. 

The Dimensional formula to express the quantity- energy is described as  [M1 L2 T-2].

SURFACE TENSION

The term surface tension can be described as the result of an asymmetric force distribution experienced at the surface of a water molecule. The quantity surface tension has a SI unit expressed as Newton per metre. The dimensional formula to express the quantity is described as [M1 T-2 ]

SURFACE ENERGY 

The extra amount of energy stored in the surface layer of the molecule is called surface energy. This extra energy is present because moving a molecule from deep inside to the surface creates an inwardly directed resultant force which then leads to storing potential energy. The dimensional formula to express the quantity is described as [M1 T-2 ]

CONCLUSION 

The term ‘dimensional formula’ refers to the expression used for representing a physical quantity in reference to a base quantity. Simply, the product of different powers is used to express a quantity in reference to base quantities. The base quantities are the prerequisites to derive the physical quantities. One letter symbols are used to represent the base quantities. Fundamental quantities, for example, length, time and mass, etc. are used to show the amount. Various physical quantities have the same unit and dimensional formula, for instance, surface tension and surface energy, impulse and linear momentum, torque and work, frequency, angular velocity and velocity gradient.