Why Does the Shell Theorem Lemma Allow Multiplication by \cos \varphi?

  • Thread starter Thread starter TMO
  • Start date Start date
  • Tags Tags
    Shell Theorem
Click For Summary
The discussion centers on the application of the shell theorem and the justification for multiplying by cos(φ) in the context of vector forces. The concern is about simplifying the expression by projecting the force contribution (dF) along the radius vector (r). The multiplication by cos(φ) is explained as a result of the dot product, which captures the component of the force in the direction of the radius vector. This projection is crucial for accurately calculating the net force in spherical coordinates. Understanding this relationship is essential for applying the shell theorem correctly.
TMO
Messages
45
Reaction score
1
320px-Shell-diag-1.png


I am concerned regarding a lemma of the shell theorem. Specifically, I am concerned with the idea that due to the vector nature of the forces, that one can simplify this:

271c7a94be4496a99e2534e2ceae7751.png


into this:

63898081c163a2fc2c132f68be1bf017.png


Could somebody precisely explain why we're allowed to multiply in the \cos \varphi in the second equation?
 
Physics news on Phys.org
It's a dot product with the radius unit vector. You're projecting the contribution of dF (which I'm inferring is along s) along the radius vector r
 

Thread 'Does acceleration affect impact energy vs constant velocity?'

I do not have a good working knowledge of physics yet. I tried to piece this together but after researching this, I couldn’t figure out the correct laws of physics to combine to develop a formula to answer this question. Ex. 1 - A moving object impacts a static object at a constant velocity. Ex. 2 - A moving object impacts a static object at the same velocity but is accelerating at the moment of impact. Assuming the mass of the objects is the same and the velocity at the moment of impact...
View full post »

Similar threads

Confusion about the electron sub shells.
  • · Replies 6 ·
Replies
6
Views
8K
Undergrad Virial Theorem and the Ergodic hypothesis
  • · Replies 2 ·
Replies
2
Views
3K
Why is it that acceleration has multiple components in SHM?
  • · Replies 5 ·
Replies
5
Views
3K
Differential Spherical Shells - Triple Integrals
  • · Replies 3 ·
Replies
3
Views
2K
Simple question on Gravito - Magnetism inside Shell
  • · Replies 13 ·
Replies
13
Views
3K
Undergrad Tensor Products - Understanding Cooperstein, Theorem 10.2
  • · Replies 1 ·
Replies
1
Views
2K
MHB Tensor Products - Basic Understanding of Cooperstein, Theorem 10.2
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad Laplacian of Retarded Potential: How to Derive the Equation Mathematically?
  • · Replies 4 ·
Replies
4
Views
2K
Direct and alternating current, a personal approach to understand them
  • · Replies 1 ·
Replies
1
Views
2K
Does Busch's Theorem Offer a Simplified Proof of Gleason's Theorem?
  • · Replies 54 ·
2
Replies
54
Views
11K