Center of mass of spherical shell inside of cone (Apostol Problem).

  • #1
zenterix
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Homework Statement
A homogenous spherical shell of radius ##a## is cut by one nappe of a right circular cone whose vertex is at the center of the sphere. If the vertex angle of the cone is ##\alpha##, where ##0<\alpha<\pi##, determine (in terms of ##a## and ##\alpha## the center of mass of the portion of the spherical shell that lies inside the cone.
Relevant Equations
##\iint_S f(x,y,z)dS##
I am asking this question because my solution does not seem to match the solution at the end of the book (Apostol Vol II, section 12.10, problem 9).

Here is my attempt to solve this problem.

If our coordinate system is chosen such that the z-axis lines up with the axis of the cone then by symmetry the ##x## and ##y## coordinates of the center of mass will both be zero.

All that is left is to calculate the z-coordinate of the center of mass.

Using spherical coordinates, the cone is ##\theta=\alpha/2## and the sphere is ##\rho=a##.

The parameterized equation for the sphere is ##\vec{r}=\langle a\sin{\theta}\cos{\phi}, a\sin{\theta}\sin{\phi}a\cos{\theta}\rangle##.

The portion inside the cone is the red area below

1698899239437.png


Just to be clear about the coordinate system, I am using spherical coordinates defined as follows

1698899257952.png


Now, the spherical shell has a constant density of ##d(x,y,z)=\frac{M}{4\pi a^2}##.

Let's define ##f(x,y,z)=z\cdot d(x,y,z)##.

Then our calculation of ##z_{cm}## is

$$z_{cm}=\frac{\int\int_S f(x,y,z) dS}{\int\int_S d(x,y,z)dS}$$

$$=\frac{\int\int_S f(\vec{r}(\theta,\phi))\cdot \lVert \frac{d\vec{r}}{d\theta}\times\frac{d\vec{r}}{d\phi} \rVert d\theta d\phi}{\int\int_S d(\vec{r}(\theta,\phi)) \lVert \frac{d\vec{r}}{d\theta}\times\frac{d\vec{r}}{d\phi} \rVert d\theta d\phi}$$

The denominator is

$$\frac{M}{4\pi a^2}\int_0^{2\pi}\int_0^{\alpha/2} a^2\sin{\theta}d\theta d\phi$$

$$=\frac{M(1-\cos{(\alpha/2)}}{2}$$

And the numerator is

$$\frac{M}{4\pi a^2}a^3\int_0^{2\pi}\int_0^{\alpha/2} \sin{\theta}\cos{\theta} d\theta d\phi$$

$$=\frac{Ma\sin^2{(\alpha/2)}}{4}$$

Thus when we put numerator and denominator together we get

$$\frac{\frac{Ma\sin^2{(\alpha/2)}}{4}}{\frac{M(1-\cos{(\alpha/2)}}{2}}$$

$$=\frac{a\sin^2{(\alpha/2)}}{2(1-\cos{(\alpha/2)})}$$

However, it seems that either I have made a mistake or the solutions manual is wrong, because they have the answer

On the axis of the cone, at a distance ##\frac{1}{4}a\frac{1-\cos{\alpha}}{1-\cos{(\alpha/2)}}## from the center of the sphere.
 
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  • #2
Are you familiar with the formula for the cosine of a double angle ##\cos(2x) = \ldots##?
 
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  • #3
Orodruin said:
Are you familiar with the formula for the cosine of a double angle ##\cos(2x) = \ldots##?
Ah yes, that is indeed the solution to my woes.

1698903446829.png
 
  • #4
I will say that your expression looks tidier. The given expression likely originated from using that ##\sin(\theta)\cos(\theta) = \sin(2\theta)/2## before integration.
 
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FAQ: Center of mass of spherical shell inside of cone (Apostol Problem).

What is the center of mass of a spherical shell inside a cone?

The center of mass of a spherical shell inside a cone is the point where the mass of the shell can be considered to be concentrated. It is determined by integrating the mass distribution of the shell over its volume, taking into account the constraints imposed by the cone.

How do you set up the coordinate system for finding the center of mass?

To set up the coordinate system, place the vertex of the cone at the origin (0,0,0) and align the cone's axis along the z-axis. The spherical shell is typically centered at the origin with a radius R. This setup simplifies the integration process by using cylindrical or spherical coordinates.

What are the key steps involved in calculating the center of mass?

The key steps include: 1) Defining the mass element (dm) of the spherical shell, 2) Expressing the coordinates of the mass element in terms of the chosen coordinate system, 3) Setting up the integral for the mass distribution within the constraints of the cone, and 4) Solving the integral to find the coordinates of the center of mass.

Why is symmetry important in solving this problem?

Symmetry is crucial because it can simplify the integration process. In this problem, the spherical shell and the cone have rotational symmetry about the z-axis. This symmetry allows us to reduce the problem to a one-dimensional integral along the z-axis, as the x and y coordinates of the center of mass will be zero due to the symmetry.

What mathematical techniques are typically used to solve this problem?

Mathematical techniques often used include setting up and evaluating integrals in spherical or cylindrical coordinates, using symmetry arguments to simplify the integrals, and applying the center of mass formula for continuous mass distributions. In some cases, numerical methods might be employed if the integrals are too complex to solve analytically.

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