Gravitation | Mass, Weight and Weightlessness

MASS

The mass of a body is the quantity of matter contained in it.

Mass is a scalar quantity.

The unit of mass is kilogram.

A body contains the same quantity of matter whether it be on the earth, moon or even in outer space.

Thus, the mass of a body is constant and does not change from place to place.

Mass of a body is usually denoted by the small ‘m’.

Mass of a body is a measure of inertia of the body and hence it is also known as inertial mass.

The mass of a body cannot be zero.

WEIGHT

We know that the earth attracts every object with a certain force and this force depends on the mass (m) of the object and the acceleration due to gravity (g).

The weight of an object is the force with which it is attracted towards the earth.

We know that $\displaystyle F=m\times a$

That is $\displaystyle F=m\times g$

The force of attraction of the earth on an object is known as the weight of the object. It is denoted by W.

So we have, $\displaystyle W=m\times g$

As the weight of an object is the force with which it is attracted towards the earth, the S.I. unit of weight is the same as that of force i.e. Newton (N).

The weight is a force acting vertically downwards; it has both magnitude and direction, so it is a vector quantity.

The value of g is constant at a given place. Therefore at a given place, the weight of an object is directly proportional to the mass, say m, of the object, that is, $\displaystyle W\,\propto \,m$ . It is due to this reason that at a given place, we can use the weight of an object as a measure of its mass.

The mass of an object remains the same everywhere, that is, on the earths and on any planet whereas its weight depends on its location.

Weight of a freely falling Body

Let us suppose a body is placed on a lift, the weighing machine will show the weight of the body n its scale. If now the lift is made to fall freely due to gravity, both the weighing machine as well as the body will fall with same acceleration i.e., with g in the downward direction. The body will, therefore, not press the weighing machine with any force and hence show zero weight. Thus a body is weightless during free fall.

Weightlessness in space

Consider an astronaut in a space ship orbiting the earth about 1000 km above its surface. At that distance from the earth, the force of gravity of earth is still quite strong. Since the acceleration due to gravity is not zero, the weight of astronauts in the space ship certainly cannot be zero. But we all have seen them on T.V., floating is a space ship and believe that in this situation they are weightless. This can be explained as follows:

Then the astronaut in the space ship is orbiting the earths, then both, the astronaut and the spaceship are in a continuing state of free fall towards the earth with the same acceleration due to gravity. Since the downward acceleration of the astronaut is the same as that of the spaceship he does not exert any force on the sides of the space ship and a weighing machine kept is the space vehicle will show his weight to be zero. Though the free fall of a body produces a feeling of weightlessness but a true weightlessness can be experienced by a spaceship is a region of outer space where the acceleration due to gravity ‘g’ is zero.

Weight of an object on the Moon

We have learnt that the weight of an object on the earth is the force with which the earth attracts the object.

In the same way, the weight of an object on the moon is the force with which the moon attracts that object.

The mass of the moon is less than that of the earth. Due to this the moon exerts lesser force of attraction on objects.

Let the mass of an object be m. Let its weight on the moon be Wm. Let the mass of the moon be Nm and its radius be Rm.

By applying the universal law of gravitation, the weight of the object on the moon will be

$\displaystyle {{W}_{m}}=G\frac{{{M}_{m}}\times m}{R_{m}^{2}}$

It the weight of the same object on the earth be $\displaystyle {{W}_{e}}$ . The mass of the earth is M and its radius is R.

Now, $\displaystyle {{W}_{e}}=G\frac{M\times m}{{{R}^{2}}}$

Mass of earth $\displaystyle =5.98\times {{10}^{24}}\,kg$

Radius of earth $\displaystyle =6.37\times {{10}^{6}}m$

Mass of moon $\displaystyle =7.36\times {{10}^{22}}kg$

Radius of moon $\displaystyle =1.74\times {{10}^{6}}m$

Substituting the values:

$\displaystyle {{W}_{m}}=G\frac{7.36\times {{10}^{22}}kg\times m}{{{(1.74\times {{10}^{6}}m)}^{2}}}$

$\displaystyle {{W}_{m}}=2.431\times {{10}^{10}}G\times m$ …(i)

And $\displaystyle {{W}_{e}}=1.474\times {{10}^{11}}G\times m$ …(ii)

Diving equation (i) by (ii), we get

$\displaystyle \frac{{{W}_{m}}}{{{W}_{e}}}=\frac{2.431\times {{10}^{10}}}{1.474\times {{10}^{11}}}$

or $\displaystyle \frac{{{W}_{m}}}{{{W}_{e}}}=0.165\,\,\approx \frac{1}{6}$

$\displaystyle \frac{Weight\,\,of\,\,the\,\,object\,\,on\,\,the\,\,moon}{Weight\,\,of\,\,the\,\,object\,\,on\,\,the\,\,earth}=\frac{1}{6}$

Weight of the object on the moon = (1/6) × its weight on the earth.

Question: Mass of an object is 10 kg. What is its weight on the earth?

Solution: Mass, $\displaystyle m=10\,\mathbf{kg}$

Acceleration due to gravity, $\displaystyle g=9.8\,\mathbf{m/}{{\mathbf{s}}^{\mathbf{2}}}$

$\displaystyle W=m\times g$

$\displaystyle W=10\times 9.8=98N$

Thus, the weight of the object is 98 N.

Question: An object weighs 10N when measured on the surface of the earth. What would be its weight when measured on the surface of the moon?

Solution: We know,

Weight of object on the moon = (1/6) × its weight on the earth.

That is $\displaystyle {{W}_{m}}=\frac{{{W}_{e}}}{6}=\frac{10}{6}N$ $\displaystyle =1.67\,N$

Thus, the weight of object on the surface of the moon would be 1.67 N.

Question: A man weighs 600N on the earth, what is its mass? If it was taken to the moon, his weight would be 100 N. What is his mass on moon? What is his accelerations due to gravity on the moon.

Solution: (i) Let is be the mass of body on earth.

$\displaystyle W=m\times g$

$\displaystyle 600=m\times 10$