Dimensions of Force

What is Force?

Force is the interaction between the bodies that occur by the cause of its change of state. It means the force is capable of changing the state of motion of a body (example: from a body at a rest position to a body under acceleration). Hence, we can describe the force as an agent that brings about change in the state of motion or sometimes deformation of the shape of the body.

Before Newton, many concepts were arising regarding force. Today, we use only the only concept of force, introduced by Newton.

Here, Newton said that “The Net Force acting upon a body is equal to its momentum rate changed over time.” Now, we use the modified version of this statement as “ The acceleration of an object as produced by a net force is always directly proportional to the magnitude of the net force, present in the same direction as the net force is inversely proportional to the object’s mass.” As we know, this statement is derived by the name Newton’s Second Law of Motion.


Formula of Force

As described in the second law, acceleration is the net force divided by the total mass. Hence the net force is net acceleration times of total mass. It can be expressed as follows.

Fnet=anetm,  where

  • a is the net acceleration of the body by which it is moving

  • F is the Net Force that acts on the body

  • m is the total mass of the body

The force has the formula as below.

Fnet = \[\frac{dp}{dt}\] = \[\frac{m \times dv}{dt}\] 

Where,

  • dt is the change in time, i.e., rate

  • dp is the change in momentum of the body

  • dv is the change in velocity of the body

As the momentum of the body is mass × velocity, we got the formula of force as the rate of change of momentum.


Dimensional Formula of Force

From the formula, stated above, we can see that force is the mass times acceleration.

We have the unit of mass as Kg and dimensions are  [M1] [L0] [T0]  unit of acceleration is m/s2  and expressed dimensionally as  [M0] [L1] [T−2], hence, we have the unit of force as  Kg×m/s2

So, it can be written dimensionally as,

\[\frac{[M^{1}][L^{0}][T^{0}] \times [M^{0}][L^{1}][T^{0}]}{[M^{0}][L^{0}][T^{2}]}\]

= \[\frac{Kg \times m}{s^{2}}\]

Hence, we get the dimensional formula for force as [M1][L1][T−2]


Derivation of The Dimensional Formula of Force

As we know that, Force = Mass × Acceleration ---- (1)

Because, acceleration = velocity × [time]–1 = [LT-1] × [T]-1

Thus, the dimensional formula of acceleration is = [LT-2] ---- (2)

On substituting the equation (2) in equation (1), we get,

Force = m × a

Or, F = [M] × [L1 T-2] = M1 L1 T-2.

This is also called the dimensional equation of force or the force dimension formula.

Therefore, the Force is represented dimensionally as M1 L1 T-2.


Dimensions of Force

As we all know that, the formula of force can be given as, F=ma

Force equals the mass times acceleration.

And the dimension of mass is mass.

The dimension of acceleration can be shown as velocity per time. Velocity can be shown as the distance per time.

Therefore, Mass, Time Distance, are easy to measure for us, so these are the basic dimensions we like to use.

Then, the dimension of force is Mass, Time, Distance per Time.

The other measurement systems like to say that force is the basic measurement, and that mass is derived from the force.


Why is a Gravitational Force Always Towards the Centre?

A force that purely depends only on the distance from the source has spherical symmetry. It means it will be the same on the surface of the sphere with that distance being the radius. As various bodies get attracted by the likewise force, they will fall towards the centre directly and equilibrate, if not they will be obstructed by the other forces (including their structural integrity), into a spherical shape.

One force, which is a non-radial direction-dependent is that of a magnetic field on a charged particle in motion. The same particle will follow a fixed helical path in general with the field direction (the plane of rotation, which is perpendicular to the field), never to move towards or away from the centre.

A central force is a type of force that points from the particle towards (away from) a fixed point the centre directly, in space, and whose magnitude depends only on the distance of the object to the centre.

Consider the central force which acts along the line joining the object to the centre. The representation can be given as follows.

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FAQs (Frequently Asked Questions)

1. Explain the Ratio of Dimensions of Impulsive Force to the Gravitational Force.

Ans: The idea behind standard units of measure depending on the conceptual categories (like ‘force’) is that dimensions and units within any such category are uniformed. Consequently, the units or dimensions of any force are the same, and the ratio is unity.


Besides, any force, whether gravitational, magnetic, or electrical, can be/contribute to an impulsive force. Therefore, within Newtonian mechanics, at least, all forces, being additive, have the same units.


However, if it is something outside of Newtonian physics, the situation could be taken as different. The problem is that, in Einstein's gravitational theory, the General Theory of Relativity, there exists no "gravitational force." In this theory, gravity affects space-time, which then affects the trajectories with no need for the concept, force.

2. Which is the Greater Force on the Gravitational and Magnetic Force?

Ans: Gravity is considered to be a weaker force to that of electromagnetism (EM). However, the amount of exerted force in a given circumstance will depend on the amount of mass and/or electric charge that is possessed by the interacting bodies.


Consider the Earth-Moon system. Both these are made of atoms which have mass and charge, but on each, there are EQUAL AMOUNTS of a positive and negative charge. So, the net EM force results in zero (within our ability to measure). Whereas all mass attracts all other masses, and so gravity is the dominant force in between the two bodies.


A hydrogen atom (OTOH) has a single proton and electron. Each of these particles has both mass and (opposite sign) charges. In such a case, the EM force of attraction is about 1040 (i.e., crores and crores) times as strong as the gravitational attraction.