Efficiently calculating the magnitude of gravitational force

In summary, the shell is divided into narrow rings and the volume and mass of each ring is calculated. An integration using Gauß' law and symmetry is then carried out to determine the gravitational force, and the use of calculus and symmetry is emphasized for simplicity.
  • #1
Richard Nash
1
0
I am reading Kolenkow and Kleppner's Classical Mechanics and they have tried to calculate the gravitational force between a uniform thin spherical shell of mass [itex]M[/itex] and a particle of mass [itex]m[/itex] located at a distance [itex]r[/itex] from the center.


The shell has been divided into narrow rings.[itex]R[/itex] has been assumed to be the radius of the shell with thickness [itex]t[/itex] ([itex]t<<R[/itex]). The ring at angle [itex]\theta[/itex] which subtends angle [itex]d\theta[/itex] has circumference [itex]2\pi R\sin\theta[/itex].The volume is $$dV=2\pi R^2t\sin \theta d\theta$$ and its mass is $$pdV=2\pi R^2t\rho\sin\theta d\theta$$

If [itex]\alpha[/itex] be the angle between the force vector and the line of centers, [itex]dF=\frac{Gm\rho dV}{r'^2}\cos\alpha [/itex] where [itex]r'[/itex] is the distance of each part of the ring from [itex]m[/itex].

Next, an integration has been carried out using $$\cos\alpha=\frac{r-R\cos\theta}{r'}$$ and $$r'=\sqrt{r'^2+R^2-2\pi R\cos\theta}$$

Question: I would like to avoid these calculations and I was wondering if there exists a better solution.
 
Science news on Phys.org
  • #2
You can use Gauß' law and symmetry. But if you want to calculate it via an integral, I don't think there is an easier way.
 
  • #3
Richard Nash said:
Question: I would like to avoid these calculations and I was wondering if there exists a better solution.
Welcome to PF!

You certainly need to use calculus. When one does this kind of calculus problem one has to use as much symmetry as possible. It seems that is achieved by dividing the sphere into rings perpendicular to the axis through the centre of the sphere and the point mass, calculating the gravity from a ring and integrating from one end to the other. If that is what they are doing, that is as simple as it gets.

AM
 

1. What is the formula for calculating gravitational force?

The formula for calculating gravitational force is F = (G * m1 * m2)/r^2, where F is the force, G is the gravitational constant (6.67 x 10^-11), m1 and m2 are the masses of the two objects, and r is the distance between the two objects.

2. How is gravitational force related to mass and distance?

Gravitational force is directly proportional to the mass of the two objects and inversely proportional to the square of the distance between them. This means that as the mass of the objects increases, the force of gravity between them also increases. However, as the distance between the objects increases, the force of gravity decreases.

3. What units are used to measure gravitational force?

The SI unit for gravitational force is Newtons (N). However, it can also be measured in terms of kilogram-meter per second squared (kg*m/s^2).

4. How does the magnitude of gravitational force change when the distance between two objects decreases?

The magnitude of gravitational force increases as the distance between two objects decreases. This is because the force of gravity is inversely proportional to the square of the distance between the objects. So, as the distance decreases, the force increases exponentially.

5. Can gravitational force be negative?

No, gravitational force cannot be negative. It is always a positive value, as it is a measure of the attraction between two objects. If the force is directed in the opposite direction, it is simply denoted with a negative sign in the equation.

Similar threads

  • Calculus and Beyond Homework Help
Replies
3
Views
488
  • Calculus
Replies
29
Views
517
  • Advanced Physics Homework Help
Replies
1
Views
301
  • Introductory Physics Homework Help
Replies
1
Views
144
  • Introductory Physics Homework Help
Replies
2
Views
549
Replies
1
Views
820
  • Introductory Physics Homework Help
2
Replies
63
Views
2K
Replies
4
Views
190
  • Classical Physics
Replies
3
Views
1K
Back
Top