Gravitation and Weightlessness - Part III

3. Difference between mass and weight
In our everyday lives we interchange the words mass and weight. In physical terms, mass and weight are two different entities. Mass is the quantity of matter in a body. It is generally denoted by the symbol m. The SI unit for measuring mass is a kilogram. Mass is a scalar quantity. Under ordinary conditions, mass of a body will remain unchanged from place to place. (Mass may change under exceptionally certain conditions[1]).

In Physics, there are two types of masses that are considered : inertial mass and gravitational mass. Inertial mass comes into picture because mass or body resists changes in its state of motion (Newtonís First Law of Motion). Thus inertial mass is defined by the equation

m = force/acceleration

The force in the above equation is that force which is causing the motion.

Gravitational mass on the other hand, is the mass of a body as measured by force of attraction by another body. Newtonís inverse squared law of gravitation defines the force.

Under uniform gravitational field, inertial mass and gravitational mass are equal. This means that when you are not moving on the earth, your inertial mass is same as your gravitational mass. In this case your gravitational mass comes from the attractive force of gravity between yourself and the earth.

Weight of a body is the force with which its mass is attracted to the earth. It is generally denoted by the symbol W.  The force is given by

W (force)  = mass (m) x acceleration due to gravity (g)

W = mg

We have seen in the earlier section that

                         M
    g   =   G        ,      M = mass of the earth, R = radius of the earth.
                         R²

Weight is a vector quantity, always pointing towards the center of the earth.  The SI unit of weight is Newtons.  

Mass of a body may remain constant but if g changes, its weight changes. We have seen in the earlier section how g changes in a spacecraft, which is at an altitude of 300 km from the surface of the earth. In the spacecraft, a person whose mass on the surface of the earth is 50 kg, will experience weightlessness; even though his mass has not changed.

Why does an objectís weight on the moon become 1/6^th of that on the earth : We have seen in example 2 in the last section that g_moon   =  1.63 m/s², which is 1/6^th of that of the earth g_earth   =  9.8 m/s². We can solve the same question from another angle.

Let m = mass of an object

Its weight on the earth is

                             M_earth
     W_earth =    G         x m
                            R²_earth

Its weight on the moon is

                             M_moon
    W_moon=     G          x m
                             R²_moon

We know that the mass of the earth M_earth = 6 x 10²⁴ kg

And the radius of the earth R_earth = 6.4 x 10³ km

Also we know that the mass of the moon M_moon = 7.3 x 10²² kg

And the radius of the moon R_moon = 1740 km

The ratio of the mass of the body on the earth versus moon is 82.1.

The ratio of the radius of the earth and the moon is 3.67.

Substituting these numbers, we get,

  W_moon              M_moon                      R²_earth               (3.67)²                1
     =              x                   =          =       
  W_earth              M_earth                        R²_moon              82.1                   6

Thus we see that if an object has the same mass m, its weight on the moon will be 1/6^th its weight on the earth.

[1] Mass and energy are related by Einsteinís famous equation E = mc²; mass may change into energy in certain nuclear reactions. Mass may change under relativistic conditions when an object moves nearly at the speed of light. But these aspects are beyond the scope of the present syllabus.

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