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

• MHB
• richatomar
In summary, the vector field F(x, y, z) is conservative and has the potential function $e^(y^2+ z)$. The sphere is heavier than the 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?

richatomar said:
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?

richatomar said:
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*}.

1. What is the moment of inertia for a solid sphere?

The moment of inertia for a solid sphere is given by the formula I = (2/5)mr^2, where m is the mass of the sphere and r is the radius.

2. How do you find the moment of inertia for a cylinder?

To find the moment of inertia for a cylinder, you can use the formula I = (1/2)mr^2, where m is the mass of the cylinder and r is the radius.

3. What is the moment of inertia for a solid cone?

The moment of inertia for a solid cone is given by the formula I = (3/10)mr^2, where m is the mass of the cone and r is the radius.

4. How does the shape of a solid affect its moment of inertia?

The shape of a solid directly affects its moment of inertia. Objects with a larger radius or greater mass will have a larger moment of inertia, while objects with a smaller radius or less mass will have a smaller moment of inertia.

5. Can the moment of inertia for a solid be negative?

No, the moment of inertia for a solid cannot be negative. It is a measure of an object's resistance to changes in rotational motion, so it must always be a positive value.

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