Problem 3 : A spherical cavity is made in a solid sphere of mass “M” and radius “R” as shown in the figure. Find the gravitational field at the center of cavity due to remaining mass.

Figure 3: The gravitational field at the center of spherical cavity

Superposition principle

Solution : According to superposition principle, gravitational field (E) due to whole mass is equal to vector sum of gravitational field due to remaining mass ( E 1 E 1 ) and removed mass ( E 2 E 2 ).

E = E 1 + E 2 E = E 1 + E 2

The gravitation field due to a uniform solid sphere is zero at its center. Therefore, gravitational field due to removed mass is zero at its center. It means that gravitational field due to solid sphere is equal to gravitational field due to remaining mass. Now, we know that “E” at the point acts towards center of sphere. As such both “E” and “ E 1 E 1 ” acts along same direction. Hence, we can use scalar form,

E 1 = E E 1 = E

Now, gravitational field due to solid sphere of radius “R” at a point “r” within the sphere is given as :

E = G M r R 3 E = G M r R 3

Here,

Figure 4: The gravitational field at the center of spherical cavity

Superposition principle

r = R − R 2 = R 2 r = R − R 2 = R 2

Thus,

⇒ E = G M R 2 R 3 = G M 2 R 2 ⇒ E = G M R 2 R 3 = G M 2 R 2

Therefore, gravitational field due to remaining mass, “ E 1 E 1 ”, is :

⇒ E 1 = E = G M 2 R 2 ⇒ E 1 = E = G M 2 R 2

Problem 4 : Two concentric spherical shells of mass “ m 1 m 1 ” and “ m 2 m 2 ” have radii “ r 1 r 1 ” and “ r 2 r 2 ” respectively, where r 2 r 2 > r 1 r 1 . Find gravitational intensity at a point, which is at a distance “r” from the common center for following situations, when it lies (i) inside smaller shell (ii) in between two shells and (iii) outside outer shell.

Solution : Three points “A”, “B” and “C” corresponding to three given situations in the question are shown in the figure :

Figure 5: The gravitational field at three different points

Superposition principle

The point inside smaller shell is also inside outer shell. The gravitational field inside a shell is zero. Hence, net gravitational field at a position inside the smaller shell is zero,

E 1 = 0 E 1 = 0

The gravitational field strength due to outer shell ( E o E o ) at a point inside is zero. On the other hand, gravitational field strength due to inner shell ( E i E i ) at a point outside is :

⇒ E i = G M r 2 ⇒ E i = G M r 2

Hence, net gravitational field at position in between two shells is :

E 2 = E i + E o = G m 1 r 2 E 2 = E i + E o = G m 1 r 2

A point outside outer shell is also outside inner shell. Hence, net field strength at a position outside outer shell is :

E 3 = E i + E o = G m 1 r 2 + G m 2 r 2 E 3 = E i + E o = G m 1 r 2 + G m 2 r 2

⇒ E 3 = G m 1 + m 2 r 2 ⇒ E 3 = G m 1 + m 2 r 2