MHB Finding Moment of Inertia for Solids: Sphere, Cylinder & Cone

[richatomar]

question 1 The vector field F(x, y, z) = 2xi + 2yey2+z j +(ey2+z + cos z) k is conservative. Find a
corresponding potential function.

* e raise to power (Y square +z)

Question 2
Consider a solid sphere of radius R, a cylindrical shell of outer radius R, inner radius a,
and height h, and a solid circular cone of radius R and mass M. All three objects have
the same constant density ρ.
(a) Find the moments of inertia about the axis of symmetry for all three objects.
(b) Suppose the objects have the same moments of inertia. Find h and M.
(c) Given the assumptions in part (b), which is heavier, the cone or the sphere?

 

[HallsofIvy]

question 1 The vector field F(x, y, z) = 2xi + 2yey2+z j +(ey2+z + cos z) k is conservative. Find acorresponding potential function.

* e raise to power (Y square +z)

Better would be e^(y^2+ z). Best would be e^{y^2+ z} using "Latex":
https://www.latex-tutorial.com/tutorials/

Now, why have you not at least tried to do this yourself? Do know what a "potential function" is? If you do then it looks pretty straight forward to me. If you have tried this, please show what you did so that we can make comments on it.

Question 2
Consider a solid sphere of radius R, a cylindrical shell of outer radius R, inner radius a,and height h, and a solid circular cone of radius R and mass M. All three objects have
the same constant density ρ.
(a) Find the moments of inertia about the axis of symmetry for all three objects.
(b) Suppose the objects have the same moments of inertia. Find h and M.
(c) Given the assumptions in part (b), which is heavier, the cone or the sphere?

Do you know what "axis of symmertry" and "moment of inertia" mean?

 

[Prove It]

question 1 The vector field F(x, y, z) = 2xi + 2yey2+z j +(ey2+z + cos z) k is conservative. Find a
corresponding potential function.

* e raise to power (Y square +z)

Question 2
Consider a solid sphere of radius R, a cylindrical shell of outer radius R, inner radius a,
and height h, and a solid circular cone of radius R and mass M. All three objects have
the same constant density ρ.
(a) Find the moments of inertia about the axis of symmetry for all three objects.
(b) Suppose the objects have the same moments of inertia. Find h and M.
(c) Given the assumptions in part (b), which is heavier, the cone or the sphere?

$\displaystyle \begin{align*} \mathbf{F} = 2\,x\,\mathbf{i} + \left( 2\,y\,\mathrm{e}^{y^2 + z} \right) \,\mathbf{j} + \left[ \mathrm{e}^{y^2 + z} + \cos{(z)} \right] \,\mathbf{k} \end{align*}$

For $\displaystyle \begin{align*} \phi \left( x, y, z \right) \end{align*}$ to be a scalar potential function of $\displaystyle \begin{align*} \mathbf{F} \end{align*}$, we require $\displaystyle \begin{align*} \nabla \phi = \mathbf{F} \end{align*}$, so $\displaystyle \begin{align*} \frac{\partial \phi}{\partial x} = F_i, \, \frac{\partial \phi}{\partial y} = F_j \end{align*}$ and $\displaystyle \begin{align*} \frac{\partial \phi}{\partial z} = F_k \end{align*}$. Thus

$\displaystyle \begin{align*} \frac{\partial \phi}{\partial x} &= 2\,x \\ \phi &= \int{ 2\,x\,\partial x } \\ \phi &= x^2 + f\left( y, z \right) \\ \\ \frac{\partial \phi}{\partial y} &= 2\,y\,\mathrm{e}^{y^2 + z} \\ \phi &= \int{ \left( 2\,y\,\mathrm{e}^{y^2 + z} \right) \,\partial y } \\ \phi &= \mathrm{e}^{y^2 + z} + g\left( x, z \right) \\ \\ \frac{\partial \phi}{\partial z} &= \mathrm{e}^{y^2 + z} + \cos{(z)} \\ \phi &= \int{ \left[ \mathrm{e}^{y^2 + z} + \cos{(z)} \right] \,\partial z } \\ \phi &= \mathrm{e}^{y^2 + z} + \sin{(z)} + h\left( x, y \right) \end{align*}$

So comparing all the pieces we can see from the three integrals (which all are ways of writing $\displaystyle \begin{align*} \phi \left( x, y, z \right) \end{align*}$, we can see that it has to have the terms $\displaystyle \begin{align*} x^2, \, \mathrm{e}^{y^2 + z} \end{align*}$ and $\displaystyle \begin{align*} \sin{(z)} \end{align*}$, thus

$\displaystyle \begin{align*} \phi \left( x, y, z \right) = x^2 + \mathrm{e}^{y^2 + z} + \sin{(z)} + C \end{align*}$.