# Gravitational field problem

1. Dec 30, 2013

### Pranav-Arora

1. The problem statement, all variables and given/known data
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)$\frac{2GM}{9R^2}+I_0$

2. Relevant equations

3. 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!

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2. Dec 30, 2013

### 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.

3. Dec 30, 2013

### Pranav-Arora

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?

4. Dec 30, 2013

### Dick

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

5. Dec 30, 2013

### Pranav-Arora

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