Dimensional Formula of Weight

In our day-to-day lives, we often interchange the terms weight and mass. However, in physics, there is an important distinction between the two. Weight is the pull of the Earth on a specified object. It depends on the distance from the centre of the Earth. However, the mass of an object does not vary with its location. The mass of a specific object is the same on Earth as it is on the moon. In this article, we will discuss some of the fundamental properties of weight and the dimensional formula of weight.

Relationship Between Weight and Gravitational Force

Generally, when an object is dropped, it accelerates towards Earth’s centre. Newton’s second law states that the specific net force on the object is responsible for its acceleration. 

When the air resistance is negligible, the net force that works on the falling object is known as the gravitational force, which in common terms is known as weight w. Weight or w is the force due to gravity which acts on the object of mass m.

Weight is a vector quantity as it has a direction. The magnitude of weight is represented as w. Galileo showed that, in the absence of air resistance, all objects fall with the same acceleration (g).

Before understanding the dimensional formula of weight, let us derive the equation of weight in the scalar form and vector form.

So, weight can be denoted in the vector form as,

→    →

w = mg

It is expressed in the scalar form as,

w = mg

Now, g = 9.80 m/s2, which acts on Earth. Hence, the weight of a 1 kg object on Earth is calculated as, 

w = mg 

= (1.00 kg) (9.80 m/s2) 

= 9.8 N

Examples of Formula of Weight

Suppose there is a person with a mass of 80 kg. The gravity on Earth is 9.8 m/s2. Thus, in order to find the weight, we will substitute the values in the above formula.

m = 80 kg

g = 9.8m/s2

w = mg

= 80 x 9.8

= 784 N

Derivation of the Dimensional Formula of Weight

Weight (W) = Mass x Acceleration—– (1)

Acceleration = Velocity x [Time]-1

Velocity is dimensionally presented as [M0 L1 T-1].

Hence, the dimensional formula of acceleration would be, 

[M0 L1 T-1] x [Time]-1= [M0 L1 T-2]— (2)

Also, the dimension of mass = [M1 L0 T0] —–(3)

So, by replacing (2) and (3) in the equation (1), we get the following,

Weight = Mass x Acceleration

Or, W = [M1 L0 T0] x [M0 L1 T-2] = [M0 L1 T-2]

Hence, the dimensional formula weight is stated as [M0 L1 T-2]

Since you have now understood the examples of the dimensional formula of weight, let us understand the link between mass and weight.

Relationship Between Mass and Weight

Weight is primarily defined as the force of gravity through which a body is pulled back to Earth or any other kind of celestial body. It is equal to the product of the mass of the object as well as the acceleration of gravity.

Now,

W = F= mg.

This is the primary equation that is applicable at all times, even if a specific object is not accelerating. So, when an object is under a free fall, there is no other external force apart from gravity that acts on it.

For example, your weight is 30 kg on Earth. Your mass on the moon would be 1/6th the weight of the Earth, i.e., 5 kg on the moon.

The mass and gravity on Earth are relatively less than that of the moon, so the body’s weight differs accordingly.

Mass and Weight Unit

In physics, each object has a specific unit that determines the standard measurement of the body.

• Mass is a physical quantity and has a unit. The SI unit of mass is kilogram, denoted by kg.

• However, weight is a vector quantity. Hence, it can be measured in Newton, denoted by N.

Mass-Gravity-Weight

We often use the terms mass, gravity, and weight interchangeably without understanding the difference. However, in terms of physics, it differs. 

• The mass of any specific object is the total matter in it at a specific point in time. 

• The weight is the total force that acts on the object. It is also the force exerted on an object due to gravity. It changes as and when the gravitational force changes.

The gravitational force that exerts on the object is directly proportional to the mass of the object. So when the object’s mass is heavy, it would exert a greater gravitational force and vice versa.

How to Convert Mass to Weight

Gravity is consistent on Earth and will be so in the coming 100 years. Hence, we can use a formula through which we can convert weight into mass and vice versa:

Force = Mass * Acceleration

Or, we can say,

F = ma

In the equation, force is equal to weight. Acceleration is the acceleration caused by the gravity “g”, which is 9.8 m/s2.

So now we can substitute the values as follows,

Weight = mass * 9.8

Conclusion

This article provides a thorough understanding of the relationship between mass, weight, and gravity. It explains the dimensional formula of weight and derivation of the same. The mass of an object is the same everywhere and is a scalar, whereas the weight is a vector and will change with surroundings.