Superposition principle for Gravitational field
Consider ‘n’ particles of masses m1 , m2 , .mn, distributed in space at positions 1 , 2 , 3 … etc, with respect to point P. The total gravitational field at a point P due to all the masses is given by the vector sum of the gravitational field due to the individual masses (Figure 6.11). This principle is known as superposition of gravitational fields.
Instead of discrete masses, if we have continuous distribution of a total mass M, then the gravitational field at a point P is calculated using the method of integration.
a) Two particles of masses m1 and m2 are placed along the x and y axes respectively at a distance ‘a’ from the origin. Calculate the gravitational field at a point P shown in figure below.
Gravitational field due to m1 at a point P is given by,
Gravitational field due to m2 at the point p is given by,
The direction of the total gravitational field is determined by the relative value of m1 and m2.
When m1 = m2 = m
(iˆ + jˆ = jˆ + iˆ as vectors obeys commutation law).
total points towards the origin of the co-ordinate system and the magnitude of total is Gm/a2.
Qualitatively indicate the gravitational field of Sun on Mercury, Earth, and Jupiter shown in figure.
Since the gravitational field decreases as distance increases, Jupiter experiences a weak gravitational field due to the Sun. Since Mercury is the nearest to the Sun, it experiences the strongest gravitational field.