What is the most efficient method for solving this gravitational field problem?

[Saitama]

Homework Statement

A thin hemispherical shell of mass M and radius R is placed as shown in figure. The magnitude of gravitational field at P due to the hemispherical shell is $I_0$. The magnitude of gravitational field at Q due to thin hemispherical shell is given by

A)$I_0/2$

B)$I_0$

C)$\frac{2GM}{9R^2}-I_0$

D)$\frac{2GM}{9R^2}+I_0$

Homework Equations

The Attempt at a Solution

I tried the problem using spherical coordinates and ended up with some messy integrals. Since this is an exam problem, I wonder if I really need to solve those integrals as it would take a lot of time. (I solved the integrals using Wolfram Alpha and the result was not nice so I immediately dropped the approach.) I believe there is a shorter way to solve this.

Any help is appreciated. Thanks!

 

[Dick]

Homework Statement

A thin hemispherical shell of mass M and radius R is placed as shown in figure. The magnitude of gravitational field at P due to the hemispherical shell is $I_0$. The magnitude of gravitational field at Q due to thin hemispherical shell is given by

A)$I_0/2$

B)$I_0$

C)$\frac{2GM}{9R^2}-I_0$

D)$\frac{2GM}{9R^2}+I_0$

Homework Equations

The Attempt at a Solution

I tried the problem using spherical coordinates and ended up with some messy integrals. Since this is an exam problem, I wonder if I really need to solve those integrals as it would take a lot of time. (I solved the integrals using Wolfram Alpha and the result was not nice so I immediately dropped the approach.) I believe there is a shorter way to solve this.

Any help is appreciated. Thanks!

There is a much faster way. You can picture the field due to the hemisphere as the sum of the field from a whole sphere and the field from an imaginary hemisphere of mass -M covering the lower half of the sphere.

 

[Saitama]

Hi Dick! :)

There is a much faster way. You can picture the field due to the hemisphere as the sum of the field from a whole sphere and the field from an imaginary hemisphere of mass -M covering the lower half of the sphere.

I consider a sphere of radius R and mass 2M.

The field at P is given by:

$$\frac{2GM}{9R^2}+E_{-M}=I_0=E_M$$

where $E_{-M}$ represents field at P due to the imaginary hemisphere of mass -M and $E_M$ represents the field at P due to hemisphere of mass M.

The field at Q is given by:

$$\frac{2GM}{9R^2}+E'_{-M}$$

where $E'_{-M}$ is the field at Q due to imaginary hemisphere of mass -M.

Since $E'_{-M}=-E_M=-I_0$, the field at Q is given by:

$$\frac{2GM}{9R^2}-I_0$$

Is this correct?

 

[Dick]

Hi Dick! :)



I consider a sphere of radius R and mass 2M.

The field at P is given by:

$$\frac{2GM}{9R^2}+E_{-M}=I_0=E_M$$

where $E_{-M}$ represents field at P due to the imaginary hemisphere of mass -M and $E_M$ represents the field at P due to hemisphere of mass M.

The field at Q is given by:

$$\frac{2GM}{9R^2}+E'_{-M}$$

where $E'_{-M}$ is the field at Q due to imaginary hemisphere of mass -M.

Since $E'_{-M}=-E_M=-I_0$, the field at Q is given by:

$$\frac{2GM}{9R^2}-I_0$$

Is this correct?

Hi Pranav-Arora! Yes, that's correct. This sort of a method is called 'using superposition'. For sort of obvious reasons.

 

[Saitama]

Hi Pranav-Arora! Yes, that's correct. This sort of a method is called 'using superposition'. For sort of obvious reasons.

Yes, I have heard of this method, thanks a lot Dick!