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Need a 2nd opinion on my solution.
A point mass moves frictionlessly in a circle inside a parabolic cup, with the radius at the top being R. The particle's position vector makes an angle theta wrt the center of symmetry (generatrix going from -z to +z).
F=ma
Confined to a certain height implies [tex]mg = N\cos \theta[/tex]
Circular motion at an arbitrary radius r<R gives
[tex]m\frac{v^2}{r}=N\sin\theta[/tex]
which yields
[tex]\frac{v^2}{rg} = tan \theta[/tex]
In my attempt to leave this as a function only of the given parameters (R,g and theta), I use the equation for a paraboloid:
[tex]z=x^2 + y^2[/tex]
and the fact that
[tex]r=sqrt(z)\\
\frac{r}{z}=\tan \theta
[/tex]
to get
[tex]v=\sqrt(g)[/tex]
This doesn't make sense to me on physical grounds. My argument: the speed of the particle in a paraboloid should depend on the angle to enable the requirement that at higher values of r<R, a greater y-component of the normal force is required to remain at that height. Also, units are off by a factor of [L]^(1/2).
Just what am I doing wrong?
Homework Statement
A point mass moves frictionlessly in a circle inside a parabolic cup, with the radius at the top being R. The particle's position vector makes an angle theta wrt the center of symmetry (generatrix going from -z to +z).
Homework Equations
F=ma
The Attempt at a Solution
Confined to a certain height implies [tex]mg = N\cos \theta[/tex]
Circular motion at an arbitrary radius r<R gives
[tex]m\frac{v^2}{r}=N\sin\theta[/tex]
which yields
[tex]\frac{v^2}{rg} = tan \theta[/tex]
In my attempt to leave this as a function only of the given parameters (R,g and theta), I use the equation for a paraboloid:
[tex]z=x^2 + y^2[/tex]
and the fact that
[tex]r=sqrt(z)\\
\frac{r}{z}=\tan \theta
[/tex]
to get
[tex]v=\sqrt(g)[/tex]
This doesn't make sense to me on physical grounds. My argument: the speed of the particle in a paraboloid should depend on the angle to enable the requirement that at higher values of r<R, a greater y-component of the normal force is required to remain at that height. Also, units are off by a factor of [L]^(1/2).
Just what am I doing wrong?
Last edited: