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Question: A uniform solid sphere of mass \(M\) and radius \(R\) is surrounded symmetrically by a uniform thin ...

A uniform solid sphere of mass MM and radius RR is surrounded symmetrically by a uniform thin spherical shell of equal mass and radius 2R2R The value of gravitational potential at a distance of 32R\dfrac{3}{2}R from the centre is :
A. −2GM3R - \dfrac{{2GM}}{{3R}}
B. −5GM6R - \dfrac{{5GM}}{{6R}}
C. −4GM3R - \dfrac{{4GM}}{{3R}}
D. −7GM6R - \dfrac{{7GM}}{{6R}}

Explanation

Solution

In order to solve this question, we will first calculate the gravitational potential due to the sphere at asked distance and then we will find potential due to the thin spherical shell and then we will add both potentials to figure out the net potential.

Formula used:
Gravitational potential due to sphere of mass MM and radius RR at a distance r>Rr > R is given by,
V=−GMrV = \dfrac{{ - GM}}{r}
Gravitational potential due to thin spherical shell having mass MM and radius RR at any distance r<Rr < R is given by
V=−GMRV = \dfrac{{ - GM}}{R}

Complete step by step answer:
According to the question we have given that the mass of the sphere is M and radius of sphere is R.Distance at which potential is need to calculate r=32Rr = \dfrac{3}{2}R so, r>Rr > R hence, gravitational potential at such distance is calculated by using the formula
V=−GMrV = \dfrac{{ - GM}}{r}
⇒V=−2GM3R→(i)\Rightarrow V = \dfrac{{ - 2GM}}{{3R}} \to (i)

Now, for thin spherical shell mass of shell is M and radius of shell is 2R2R.Distance at which potential is need to Calculate r=32Rr = \dfrac{3}{2}R so, r>2Rr > 2R hence, gravitational potential at such distance is calculated by using the formula
V=−GM2RV = \dfrac{{ - GM}}{{2R}}
⇒V=−GM2R→(ii)\Rightarrow V = \dfrac{{ - GM}}{{2R}} \to (ii)
So, net potential at distance of 32R\dfrac{3}{2}R is sum of equation (i) and (ii) we get,
Vnet=−2GM3R+−GM2R{V_{net}} = \dfrac{{ - 2GM}}{{3R}} + \dfrac{{ - GM}}{{2R}}
⇒Vnet=−4GM−3GM6R\Rightarrow {V_{net}} = \dfrac{{ - 4GM - 3GM}}{{6R}}
∴Vnet=−7GM6R\therefore {V_{net}} = \dfrac{{ - 7GM}}{{6R}}

Hence, the correct option is D.

Note: It should be remembered that, gravitational potential is a scalar quantity so, it can be added simply using scalar law of addition. Also, the gravitational potential at any point inside the spherical shell is always constant while for a solid sphere it depends upon the mass distribution of the sphere.