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Question
A solid sphere of uniform density and radius 4 unit is located with its center at the origin O of co-ordinates. Two spheres of equal radii 1 unit, with their centresat A (-2,0,0) and B (2, 0,0) respectively, are taken out of the solid leaving behind spherical cavities as shown in figure. Then
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Solution
The correct options are
A The gravitational field due to this object at the origin is zero.
C The gravitational potential is same at all points on the circle Y2+z2=36
D The gravitational potential is same at all points on the circle Y2+z2=4
a) Due to solid sphere of uniform density, gravitational field is zero at centre O. The cavities at A and B can be treated as negative masses. The cavities are situated on opposite sides of the centre O. The gravitational forces, exerted by the cavity - masses, on a mass at O are opposite. Hence the resultant force on mass at O is zero. Thus the gravitational force due to this object at the origin O is zero, option (a) is correct (b) option (b) is incorrect in view of above discussion. (c) and (d) - These are correct options. Consider the circle, y2+z2=36. The centre of circle is (0,0,0). The radius of circle is 6 units.
The circle lies in (y-z) plane. It is JL to x-axis. For a point situated on or outside the sphere, the mass of sphere can be assumed to be situated at the centre. All the points of circle y2+z2=36 are equidistant from the centre O of the sphere, where the mass is supposed to be concentrated. Hence the gravitational potential is the same at all points of circle y2+z2=36 Option (c) is correct.
Consider the circle y2+z2=4.
It centre lies at (0,0,0). Its radius is 2 units. It lies in y-Tf plane, perpendicular to x-axis A discussion on the lines of option (c) leads to the conclusion that (d) is a correct option.
Thus (c) and (d) represent correct options.
A The gravitational field due to this object at the origin is zero.
C The gravitational potential is same at all points on the circle Y2+z2=36
D The gravitational potential is same at all points on the circle Y2+z2=4
a) Due to solid sphere of uniform density, gravitational field is zero at centre O. The cavities at A and B can be treated as negative masses. The cavities are situated on opposite sides of the centre O. The gravitational forces, exerted by the cavity - masses, on a mass at O are opposite. Hence the resultant force on mass at O is zero. Thus the gravitational force due to this object at the origin O is zero, option (a) is correct (b) option (b) is incorrect in view of above discussion. (c) and (d) - These are correct options. Consider the circle, y2+z2=36. The centre of circle is (0,0,0). The radius of circle is 6 units.
The circle lies in (y-z) plane. It is JL to x-axis. For a point situated on or outside the sphere, the mass of sphere can be assumed to be situated at the centre. All the points of circle y2+z2=36 are equidistant from the centre O of the sphere, where the mass is supposed to be concentrated. Hence the gravitational potential is the same at all points of circle y2+z2=36 Option (c) is correct.
Consider the circle y2+z2=4.
It centre lies at (0,0,0). Its radius is 2 units. It lies in y-Tf plane, perpendicular to x-axis A discussion on the lines of option (c) leads to the conclusion that (d) is a correct option.
Thus (c) and (d) represent correct options.
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