Rod Kinetic Energy in Spherical Coordinates

Consider the following mapping from Cartesian coordinates to spherical coordinates:

x = r c o s θ s i n ϕ y = r s i n θ s i n ϕ z = r c o s ϕ x = r \, cos \theta \, sin \phi \\ y = r \, sin \theta \, sin \phi \\ z = r \, cos \phi

In this coordinate system, θ \theta is the angle in the x y x y plane with respect to the + x +x axis. ϕ \phi is the angle with respect to the + z +z axis.

Consider a uniform rigid rod of mass M M and length L L , with one end hinged at the origin, and the other end free to move. The hinge permits any combination of ( θ , ϕ ) (\theta, \phi) , subject to the natural constraints on each parameter.

The kinetic energy of the rod can be expressed as follows:

E = β M L 2 ( ϕ ˙ 2 + s i n 2 ϕ θ ˙ 2 ) E = \beta \, M L^2 \Big (\dot{\phi}^2 + sin^2 \phi \,\, \dot{\theta}^2 \Big )

Given that β \beta is a positive real number, what is its value?


The answer is 0.1666667.

This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try refreshing the page, (b) enabling javascript if it is disabled on your browser and, finally, (c) loading the non-javascript version of this page . We're sorry about the hassle.

1 solution

Vitor Juiz
Apr 26, 2018

0 pending reports

×

Problem Loading...

Note Loading...

Set Loading...