What is the relationship between pressure and force inside a spherical object?

In summary, the man in the video is applying the formula for the gravitational force on a spherical shell to a point-sized mass, which incorrectly assumes that the net gravitational force is zero.
  • #1
PSN03
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9
Homework Statement
A uniform sphere has a mass M and radius R. Find the pressure p inside the sphere caused by gravitational compression, as a function of distance r from its centre.
Relevant Equations
Pressure*Area=Force
Force=GM1M2/R²
So I already have a solution available to this problem and the link for the solution is:


I have understood everything in the video except the part where they are equating the force
dF=GM/r²*dm
According to my reasoning the inner part of the sphere (the part below the dm element we have taken) will have mass M and can be treated as a point sized object. This object will have a field E=GM/r² at a distance of r from the centre.
The dm element that we are taking into consideration is a spherical shell. It's centre of mass shoul coincide with the centre of mass of the sphere, hence there shouldn't be any distance between the two bodies to apply Newton's law of gravitation.
So my doubt it howare they applying the formula to dm mass even though it isn't point sized and distributed over a surface.
 
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  • #2
PSN03 said:
I have understood everything in the video except the part where they are equating the force
dF=GM/r²*dm
This expression for the gravitational force on the spherical shell can be misleading. The net gravitational force on the shell is zero as you can easily see by considering diametrically opposite elements of mass in the shell:
 
  • #3
TSny said:
This expression for the gravitational force on the spherical shell can be misleading. The net gravitational force on the shell is zero as you can easily see by considering diametrically opposite elements of mass in the shell:
Yes you have a point there. Then how is the man in video applying it ?
Moreover we should consider the centre of mass of the elemental sphere while writing the above expression...right?
 
  • #4
PSN03 said:
I have understood everything in the video except the part where they are equating the force
dF=GM/r²*dm
This expression for the gravitational force on the spherical shell isn't really correct. The net gravitational force on the shell is zero as you can easily see by considering diametrically opposite elements of equal mass in the shell:
1597851344568.png


The two gravitational forces shown add to zero. Thus, the net gravitational force on the entire shell is zero.

It is less confusing if you consider just one element of mass of the shell as shown below:

1597851861337.png


Consider the balance of gravitational force and pressure force on this one element.
 
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  • #5
TSny said:
This expression for the gravitational force on the spherical shell isn't really correct. The net gravitational force on the shell is zero as you can easily see by considering diametrically opposite elements of equal mass in the shell:
View attachment 267968

The two gravitational forces shown add to zero. Thus, the net gravitational force on the entire shell is zero.

It is less confusing if you consider just one element of mass of the shell as shown below:

View attachment 267971

Consider the balance of gravitational force and pressure force on this one element.
How to proceed further after taking one element of this mass shell?
 
  • #6
TSny said:
This expression for the gravitational force on the spherical shell isn't really correct. The net gravitational force on the shell is zero as you can easily see by considering diametrically opposite elements of equal mass in the shell:
View attachment 267968

The two gravitational forces shown add to zero. Thus, the net gravitational force on the entire shell is zero.

It is less confusing if you consider just one element of mass of the shell as shown below:

View attachment 267971

Consider the balance of gravitational force and pressure force on this one element.
Hey @TSny I have got what you were trying to tell. Also I found a similar video solution to this problem but it's in my regional language so I can't upload it but I have got the solution.
Thanks for your help.
Good day and stay safe😊
 
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FAQ: What is the relationship between pressure and force inside a spherical object?

What is pressure and force inside a spherical object?

Pressure and force inside a spherical object refer to the amount of force exerted on the walls of the object and the amount of force applied on the object's surface, respectively.

What is the relationship between pressure and force inside a spherical object?

The relationship between pressure and force inside a spherical object is direct and proportional. This means that as pressure increases, force also increases, and vice versa.

How is pressure and force inside a spherical object measured?

Pressure is typically measured in units of force per unit area, such as pounds per square inch (psi) or pascals (Pa). Force is measured in units of Newtons (N) or pounds (lbs). These measurements can be obtained using specialized instruments, such as pressure gauges and force sensors.

What factors can affect the pressure and force inside a spherical object?

Several factors can affect the pressure and force inside a spherical object, including the size and shape of the object, the material it is made of, and the amount of force being applied to the object's surface.

How does the pressure and force inside a spherical object affect its stability?

The pressure and force inside a spherical object play a crucial role in determining its stability. If the pressure and force are evenly distributed, the object will be stable. However, if there is an imbalance in pressure or force, the object may become unstable and potentially collapse or deform.

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