Weightlessness
Have you seen the pictures of astronauts and objects floating in satellites orbiting the Earth? This is because the acceleration due to gravity, g is small there.
This reasoning is wrong. We show a pumpkin hanging from a spring balance, which is attached to the roof of the elevator. There is also a girl standing on the floor.
Let m g be the magnitude of the gravitational pull on the pumpkin and W the force on it due to the spring. In the first case the acceleration a of the pumpkin is zero and hence
m g - W = 0
The spring scale reads W. If the elevator acquires an acceleration a, the free body diagram of the pumpkin suggests that
m g - W = m a
or the spring balance would read
W = m g - m a
If the elevator's suspension cable snaps, the elevator and all objects inside it have an acceleration g and
W = m g - m g = 0
The spring balance reads zero. The girl is amused. In fact, the normal force (reaction) N on the girl is also zero. The scale reading would also be zero if the girl were standing on a weighing scale.
The earth continues to exert force m g and M g on the pumpkin and the girl respectively. The apparent weight of the pumpkin is the reading of the spring balance and of the girl is the normal reaction of the floor. In free fall both W and N are zero. This explains apparent weightlessness.
The satellite under the influence of gravity "falls" out of its straight-line path towards the Earth |
A similar situation prevails in the case of the orbiting satellite. The satellite and all objects within it, including the astronauts experience the attractive force of Earth's gravitation and are indeed in a 'free fall'.
The force of Earth's gravity causes the satellite to deviate from its natural straight-line path and 'fall' towards the Earth. Its large horizontal velocity enables it to cross 'over the horizon' and move in a circular orbit. If the satellite is suddenly stopped and released, it will come crashing to Earth in a straight-line path. Free fall means acceleration is g.
It is interesting to live inside an orbiting satellite. One can lift a 1000 kg mass, overturn a cup of tea without spilling it to the floor and 'float' across the rooms in fairy tale fashion.
Gravitational and Inertial Mass
Gravitational Mass
Gravitational mass is the mass of the body, which determines the magnitude of gravitational pull between the body and the earth.
Let F be the gravitational force between the body of mass m and earth.
Then F = G
Where M is the mass of the earth and R is the radius of the earth.
Therefore m =
The mass of a body determined in this way is the gravitational mass of the body. The inertia of a body has no role to play in this case.
Inertial Mass
Inertial mass of body is a measure of the ability of a body to oppose the production of acceleration in it by an external force. In other words, inertial mass of a body is measure of the inertia of a body.
Consider a body of mass m lying on a smooth horizontal surface. The body tends to remain in a state of rest due to property of 'inertia of rest.' If the body is to be set in motion, some force is required to overcome this inertia. Let a force F be applied horizontally on the body such that acceleration 'a' is produced in the body.
Now, F = ma (Newton's 2^{nd} law)
Or m =
The ration of the force applied on a body placed on a horizontal frictionless surface to the acceleration produced in it is called the inertial mass. Gravity has no effect on the inertial mass of a body.
Following are the characteristics of the inertial mass.
- Inertial masses can be added, irrespective of the material involved.
- Inertial mass of a body is directly proportional to the quantity of matter contained in the body.
- Even if a body changes its state from solid to liquid or from liquid to gaseous, the inertial mass remains unaffected.
- When two bodies combine physically or chemically, the inertial mass remains unaffected.
- Inertial mass does not depend upon the temperature of the body.
- Inertial mass of a body does not depend upon the temperature of the body.
- Inertial mass of a body does not depend on the presence or absence of other bodies near it.
- When the velocity of a body approaches the velocity of light, the inertial mass of given by
m =
where m_{0} is the rest mass of a body, v is the velocity of the body and c is the velocity of light in vacuum.
Comparison
- Both are scalar quantities.
- Both are directly proportional to the quantity of matter contained in a body.
- Both are measured in the same units.
- Both are conserved in a chemical reaction.
- Both do not depend on the state or shape of the body.
- Both are equivalent to each other.
Equivalence Of Inertial And Gravitational Masses
Consider two bodies P and Q of gravitational masses m_{g} and m_{g}_{ } respectively. Let the distance of both the bodies from the centre of earth be r. According to Newton's law of gravitation, the gravitational force exerted by earth on P is given by
F = G
F = G
where M_{g} is the gravitational mass of earth. The gravitational force exerted by earth on Q is given by
F = G
=
Let m_{i} and m_{i}' be the inertial masses of P and O respectively. Suppose both the bodies are allowed to fall in vacuum from the same place near the surface of earth. If g be the value of acceleration due to gravity, then
F = m_{i }g and F' = m'_{i } g
Dividing, =
=
=
It follows from here that inertial and gravitational masses are proportional to one another. Newton was the first to devise an experiment to test directly the apparent equivalence of inertial and gravitational masses. Newton failed to detect any difference between inertial and gravitational masses. So, Newton concluded that inertial and gravitational masses are equivalent.
In 1909, Eotvos devised an apparatus, which could detect a difference of 5 parts in 10^{9} in gravitational force. He found that equal inertial masses always experience equal gravitational forces. A refined version of the Eotvos experiment was reported in 1964 by R. H. Dicke and his collaborators, who improved the accuracy of the original experiment by a factor of several hundred.
In classical physics, the equivalence of gravitational and inertial mass was looked upon as a remarkable accident having no deep significance. But Einstein realized the significance of this equivalence. He made this fact of equivalence one of the basic ideas of his general theory of relativity, which is the modern theory of gravitation.