A hollow symmetric body of height \(h\) (a surface of revolution) is formed by rotating the curve \(R(z)\) about the \(z\) axis, where \(R^2 = x^2+y^2\) is the radius in cylindrical coordinates. The endpoints of the curve satisfy \(R(h) = R_h\) and \(R(0) = 0\).
- Write down an expression for the area element \(\d A\) of a circular annulus at coordinates \((R,z)\) and a mass element \(\d m\) contained in the area \(\d A\).
- Write down an expression for the moment of inertia \(I_{zz}\) about the \(\zhat\) axis.
- Simplify \(I_{zz}\) into a single integral over \(z\) and re-interpret \(I_{zz}\) as a function:
\(\)I_{zz}[R(z)] = \int_0^h f(R,R') \d z\(\)
and determine \(f(R,R')\). Without using equations, argue that \(R(z)\) must be a minimum rather than maximum of the functional.
- Show that, since \(f(R,R')\) is independent of \(z\), we can write
\(\)\frac{\d f}{\d z} = \frac{\partial f}{\partial R}R' + \frac{\partial f}{\partial R'}R" = \frac{\d}{\d z}\left(R' \frac{\partial f}{\partial R'}\right)\(\)
and thus
\(\)f - R'\frac{\partial f}{\partial R'} = C\(\)
for some constant \(C\).
- Write down, but do not solve, the first order differential equation for \(R(z)\) in this case.