Dimensional Formula of Newton

Force is one of the most fundamental components that exists in our universe. Force is everywhere, in between the celestial bodies, like stars and planets, and in between the very small subatomic particles like electrons, protons, etc. Newton is the unit of Force. Newton’s 2nd law can be defined as an external influence that changes or attempts to change the motion of a body. Although force can be classified into various subclasses on the basis of its type and strength, conceptually, it can be imagined as either pull or push type in nature. Various forces exist in nature, for example, gravitational force, magnetic force, nuclear force, etc. In these notes of the dimensional formula of Newton, we will learn how to deduce the dimensional formula of force, meaning of momentum, and other related questions.   

Force

Force is defined as the push or pull an object exerts on another. When two items come into contact with each other, they both exert a force on each other. Stretch and squeeze are other words that can be used to describe force. What does it mean to have “force” in physics?

“Push or pull that alters the velocity at which anything with mass travels.”

You can find out how much force is at work by utilising a spring balance. Newton (N) is the SI unit of force.

Dimensional analysis

The dimensional analysis uses a set of units to determine the form of an equation, or more commonly, to ensure that the result of a computation is correct as a safeguard against many common errors.

Dimensions in units and measurements

The dimensions can be written as the powers of the fundamental units of length, mass, and time. It depicts their nature and does not show their magnitude. 

Example to write dimensions:

Let’s take the formula of the area of a rectangle.

Area of the rectangle = length x breadth

= [L1] X [L1]   (where breadth is also showing the length of the side)

= [L2]

Here, we can see length raised to the power of 2 and we cannot find the dimension of mass and time.

Hence, the dimension of the area of a rectangle is written as [M0 L2 T0]

Dimensional formula

The dimensional formula depicts the dependency of a physical quantity with fundamental physical quantity, along with the powers.

Example

Let’s take the formula of speed

Speed = Distance / Time

The distance can be written in length [L]

Time can be written as [T]

The dimensional formula would be [ M0 L1 T-1]

Hence, we can conclude that speed is only dependent on length and time, not on mass.

Dimensional analysis of Newton

As mentioned above, it is a unit of force and force can be defined as the product of mass and acceleration. 

So, F(N) = mass x acceleration

Dimension of newton = Dimension of mass x Dimension of acceleration 

Dimension of mass = [M]

Dimension of acceleration (It is the rate of change of velocity with time)

      = [velocity]/[time]

      = [displacement]/[time×time]

      = L/T×T

      = LT-2

So, the dimension of Newton = [MLT-2]

Now, we can say that the dimensional formula of newton is [MLT-2]

Importance of dimensional formula of force

• The dimensional formula of force helps us understand the physical correctness of any equation involving force in it.
• It also helps us understand the relationship between different physical quantities involving force.
• It helps us in converting units from one physical quantity to another.
• In any relationship, the constant dimensions can be found using this analysis.

Uses of the dimensional formula

1. It can prove to be a helpful tool to check the consistency and coherency of your dimensional equation.
2. The dimensional formula will be used to establish the correlation between the physical quantities of physical phenomena.
3. These formulas can be used as a device to change the units from one system to another.

Limitations of the dimensional formula

• It is not concerned with the dimensional constant.
• The formula that contains functions like  trigonometric, exponential, logarithmic, and the like cannot be derived.
• In the context of a physical quantity being scalar or vector, it does not provide any information about whether the quantity is physical or not.

Momentum

The meaning of momentum can be described in various ways, but in simple words, it gives the idea of motion of a body. It is defined as the product of mass and the velocity of the body. Momentum can be defined for both linear as well as circular motion. If a body is at rest, the momentum will be zero.  

Momentum = mass × velocity 

Conclusion

In these notes of the dimensional formula of Newton, we learnt the derivation of the dimensional formula of Newton and the meaning of momentum. With the help of the dimension of force, we can deduce dimensions of other quantities, such as momentum etc. Momentum is also a very useful concept having various implementations. For example, using the conservation of linear momentum law we can find the velocity of a body, given the other required quantities are known.