Answer
Verified
417.9k+ views
Hint: To solve this question, we need to find out the gravitational field at the surface and at a point at the depth $ R $ below the surface. Equating these, we will get the relation between the densities of the core and of the outer shell, from which the required ratio can be calculated.
Complete step-by-step solution
Consider the core and the outer shell of the planet as shown in the figure below.
Consider the point A on the surface of the planet, and the point B at a depth of $ R $ below the surface.
The acceleration due to gravity at the point A will be the sum of the acceleration due to gravity due to the outer shell and that due to the core, that is,
$ {g_A} = \dfrac{{G{M_1}}}{{{{\left( {2R} \right)}^2}}} + \dfrac{{G{M_2}}}{{{{\left( {2R} \right)}^2}}} $
$ {g_A} = \dfrac{{G{M_1} + G{M_2}}}{{4{R^2}}} $ .....................(1)
Now, the acceleration due to gravity at point B will be due to the core only since we know that the gravitational field inside a hollow spherical shell is equal to zero. So we have
$ {g_B} = \dfrac{{G{M_1}}}{{{R^2}}} $ .....................(2)
According to the question, the acceleration due to gravity at the surface of the planet is the same as that at the depth $ R $ . So we have
$ {g_A} = {g_B} $
From (1) and (2)
$ \dfrac{{G{M_1} + G{M_2}}}{{4{R^2}}} = \dfrac{{G{M_1}}}{{{R^2}}} $
Cancelling $ \dfrac{G}{{{R^2}}} $ from both the sides, we have
$ \dfrac{{{M_1} + {M_2}}}{4} = {M_1} $
$ \Rightarrow {M_2} = 3{M_1} $ .....................(3)
Now, since the density of the core is given to be $ {\rho _1} $ and is radius is equal to $ R $ , so its mass can be written as
$ {M_1} = {\rho _1} \times \dfrac{4}{3}\pi {R^3} $ .....................(4)
Similarly, we can write the mass of the outer shell as
$ {M_2} = {\rho _2} \times \left( {\dfrac{4}{3}\pi {{\left( {2R} \right)}^3} - \dfrac{4}{3}\pi {R^3}} \right) $
$ \Rightarrow {M_2} = \dfrac{4}{3}\pi {R^3}\left( {7{\rho _2}} \right) $ .....................(5)
Putting (4) and (5) in (3) we get
$ \dfrac{4}{3}\pi {R^3}\left( {7{\rho _2}} \right) = 3{\rho _1} \times \dfrac{4}{3}\pi {R^3} $
Cancelling $ \dfrac{4}{3}\pi {R^3} $ from both the sides, we get
$ 7{\rho _2} = 3{\rho _1} $
$ \dfrac{{{\rho _1}}}{{{\rho _2}}} = \frac{7}{3} $
Thus, the ratio of $ \dfrac{{{\rho _1}}}{{{\rho _2}}} $ is equal to $ 7/3 $ .
Hence, the correct answer is option A.
Note
Do not consider the mass of the outer shell to be the mass of the whole of the bigger sphere of radius $ 2R $ . This is because the bigger sphere is composed of both the outer shell and the core. The outer shell is only the region outside the core.
Complete step-by-step solution
Consider the core and the outer shell of the planet as shown in the figure below.
Consider the point A on the surface of the planet, and the point B at a depth of $ R $ below the surface.
The acceleration due to gravity at the point A will be the sum of the acceleration due to gravity due to the outer shell and that due to the core, that is,
$ {g_A} = \dfrac{{G{M_1}}}{{{{\left( {2R} \right)}^2}}} + \dfrac{{G{M_2}}}{{{{\left( {2R} \right)}^2}}} $
$ {g_A} = \dfrac{{G{M_1} + G{M_2}}}{{4{R^2}}} $ .....................(1)
Now, the acceleration due to gravity at point B will be due to the core only since we know that the gravitational field inside a hollow spherical shell is equal to zero. So we have
$ {g_B} = \dfrac{{G{M_1}}}{{{R^2}}} $ .....................(2)
According to the question, the acceleration due to gravity at the surface of the planet is the same as that at the depth $ R $ . So we have
$ {g_A} = {g_B} $
From (1) and (2)
$ \dfrac{{G{M_1} + G{M_2}}}{{4{R^2}}} = \dfrac{{G{M_1}}}{{{R^2}}} $
Cancelling $ \dfrac{G}{{{R^2}}} $ from both the sides, we have
$ \dfrac{{{M_1} + {M_2}}}{4} = {M_1} $
$ \Rightarrow {M_2} = 3{M_1} $ .....................(3)
Now, since the density of the core is given to be $ {\rho _1} $ and is radius is equal to $ R $ , so its mass can be written as
$ {M_1} = {\rho _1} \times \dfrac{4}{3}\pi {R^3} $ .....................(4)
Similarly, we can write the mass of the outer shell as
$ {M_2} = {\rho _2} \times \left( {\dfrac{4}{3}\pi {{\left( {2R} \right)}^3} - \dfrac{4}{3}\pi {R^3}} \right) $
$ \Rightarrow {M_2} = \dfrac{4}{3}\pi {R^3}\left( {7{\rho _2}} \right) $ .....................(5)
Putting (4) and (5) in (3) we get
$ \dfrac{4}{3}\pi {R^3}\left( {7{\rho _2}} \right) = 3{\rho _1} \times \dfrac{4}{3}\pi {R^3} $
Cancelling $ \dfrac{4}{3}\pi {R^3} $ from both the sides, we get
$ 7{\rho _2} = 3{\rho _1} $
$ \dfrac{{{\rho _1}}}{{{\rho _2}}} = \frac{7}{3} $
Thus, the ratio of $ \dfrac{{{\rho _1}}}{{{\rho _2}}} $ is equal to $ 7/3 $ .
Hence, the correct answer is option A.
Note
Do not consider the mass of the outer shell to be the mass of the whole of the bigger sphere of radius $ 2R $ . This is because the bigger sphere is composed of both the outer shell and the core. The outer shell is only the region outside the core.
Recently Updated Pages
How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE
Mark and label the given geoinformation on the outline class 11 social science CBSE
When people say No pun intended what does that mea class 8 english CBSE
Name the states which share their boundary with Indias class 9 social science CBSE
Give an account of the Northern Plains of India class 9 social science CBSE
Change the following sentences into negative and interrogative class 10 english CBSE
Trending doubts
Which are the Top 10 Largest Countries of the World?
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
Difference Between Plant Cell and Animal Cell
Give 10 examples for herbs , shrubs , climbers , creepers
Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
How do you graph the function fx 4x class 9 maths CBSE
Write a letter to the principal requesting him to grant class 10 english CBSE