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I have a hemispherical bowl in which I roll a small particle around the edge, starting from the top at point A with a velocity $v_o$. It travels halfway around the sphere and reaches point B, which is a vertical distance h below A, with a velocity $v_f$. Point A is a radial distance of $r_o$ from the vertical centerline and point B is a radial distance of $r$ from the vertical centerline. There is no friction. The goal is to solve for the angle, $\theta$, between the horizontal and the velocity $v_f$.

Here is a diagram of the problem scenario:

diagram of sphere

My solution relies on the assumption that angular momentum only relies on the velocities in the plane perpendicular to the vertical centerline. Is that a safe assumption? Also, when dealing with energies, is rotational KE and linear KE the same? Should I be taking RKE into account?


$$ L_o=L_f $$ $$ mr_ov_o=mrv_f\cos \theta $$ $$ \theta = \arccos(\dfrac {mr_ov_o}{mrv_f}) = \arccos(\dfrac {r_ov_o}{rv_f}) $$


$$ KE_o + PE_o = KE_f $$ $$ \frac 12 mv_o^2 + mgh = \frac 12 mv_f^2 $$ $$ v_o^2 + 2gh = v_f^2 $$ $$ \sqrt {v_o^2 + 2gh} = v_f $$


$$ \theta = \arccos(\dfrac {r_ov_o}{rv_f}) = \arccos(\dfrac {r_ov_o}{r\sqrt {v_o^2 + 2gh}}) $$

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Nice problem... –  ja72 Dec 6 '13 at 2:34
    
I do not think the mass moment if inertia is intended to be included since there is no mention of $I$ in the statement. So no on the rotational kinetic energy. Just $KE = \frac{1}{2} m \vec{v}^\top \vec{v}$. –  ja72 Dec 6 '13 at 5:25
    
@MaximUmansky: The spherical nature of the bowl would be encoded in the dependence between $h$, $r$ and $r_0$. As for the sliding versus rolling the problem says: "I roll...", the no friction part may simply mean the mechanical energy is conserved. –  user23660 Dec 6 '13 at 5:41
    
Wait a minute, the problem is over-constrainted. Two constraints are needed to define location on the surface and for point B three constraints are given. The distance $r$, the drop $h$ and the azimuthal position. This can only be true if $r = \sqrt{r_0^2-h^2}$. –  ja72 Dec 6 '13 at 5:41
    
If there is no friction then the particle would not roll, it will just slide. And if it does roll then we don't know it's shape so cannot say about the moment of inertia. So probably the rotational energy is not meant to be relevant here. But in this solution, where do we use that it is spherical surface? The equations written would be correct for any axisymmetric surface. Where do we use that the particle traveled halfway around the sphere - does it mean that the azimuthal angle traveled is Pi? Also, is the motion going to be periodic here - so the particle will return to the starting point? –  Maxim Umansky Dec 6 '13 at 5:56

3 Answers 3

Your solution looks fine to me.

Yes: the angular momentum is preserved in the horizontal plane (the weight is vertical and the reaction of the sphere surface is a central force) so your first relation is fine, just remember that $\theta$ is not the vertical angle, but lies on the plane tangent to the sphere at point B.

There are two kinds of rotational energy: the one of the particle spinning on itself which would require the particle's mass distribution to be computed, which is not given so I assume it should be neglected. What may be confusing is the particle's rotational energy around the centre of the sphere: $$E=\frac{1}{2}I\omega^2$$ For a point like mass the moment of inertia is: $$I = mr^2$$ Remembering the relation between angular and tangential velocity: $$\omega= \frac{v}{r}$$ we get exactely: $$E=\frac{1}{2}mv^2$$ In fact for a point like mass it is exactly the same to consider the kinetic energy related to the tangent velocity, or the rotational kinetic energy related to the angular velocity, for this problem I approve your choice for the first one.

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I do not have a solution, just some steps to get there.

I have parametrized the problem with spherical coordinates, $\varphi$ is the azimuthal angle (around the hoop), $\psi$ is the nutation angle (drop from horizontal plane) for a position vector

$$ \vec{r} = \begin{pmatrix} r \cos \varphi \cos \psi \\ -r \sin \psi \\ -r \sin \varphi \cos \psi \end{pmatrix} $$

The derivate gives us velocity

$$ \vec{v} = \begin{pmatrix} -r \dot\psi \cos\varphi \sin\psi - r \dot\varphi \sin\varphi \cos\psi \\ -r \dot\psi \cos \psi \\ r \dot\psi \sin\varphi \sin\psi - r \dot\varphi \cos\varphi \cos\psi \end{pmatrix} $$

The energies are

$$ \begin{aligned} PE & = -m g r \sin \psi \\ KE & = \frac{1}{2} m r^2 \left( {\dot\varphi}^2 \cos^2 \psi + {\dot\psi}^2 \right) \end{aligned} $$

At point A we know that $\varphi_0=0$, $\psi_0=0$, $\dot\varphi_0 =\frac{ v_0}{r_0}$ and $\dot\psi=0$.

At point B we know that $\varphi_1=\pi$, $\psi_1=\sin^{-1} \left(\frac{h}{r}\right)$, and that $r=\sqrt{r_0^2-h^2}$ in order for the object to remain on the surface.

Now the angle $\theta$ is kinda tricky to derive and my best guess is

$$ \begin{aligned} \tan \theta & = \frac{-\vec{v}_y}{\sqrt{\vec{v}_x^2+\vec{v}_z^2}} \\ & = \frac{\dot\varphi \cos\psi}{\sqrt{\left({\dot\varphi}^2-{\dot\psi}^2\right)\cos^2\psi + {\dot\psi}^2}} \end{aligned} $$

Now when you have $PE_0 + KE_0 = PE_1 + KE_1$ there are to variables that you need for $\theta$ ($\dot\varphi$ and $\dot\psi$) and only 1 equation. The 2nd equation must come from the angular momentum conservation, so I think you are on the right path.

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How is this an answer to any of the question about angular momentum or about rotational energy? –  user23660 Dec 6 '13 at 6:34
    
Oh yeah, the question. I forgot about it :-). –  ja72 Dec 6 '13 at 6:37

First, you are right: in this problem angular momentum around the vertical axis is conserved. This is because all forces acting on the particle have no azimuthal component.

Note, that even if we assume that the particle is rolling on the bowl surface without slippage, the angular momentum of its own rotation would be of negligible compared with the angular momentum of its movement around the bowl axis.

Second, for neglecting the rotational kinetic energy, there are ambiguities in the formulation of the problem. You wrote: "I roll a small particle". To my mind its meaning is that the particle would be rolling in a bowl, without slipping. The phrase "There is no friction" thus should mean there is no rolling friction, that is, mechanical energy is conserved. However, one may also prioritize the phrase "There is no friction" and assume that the "I roll" part to be imprecise way of saying that the particle will be sliding, without rotating. The first interpretation means that we have to make an assumption about the structure of a particle (or at least its tensor of inertia). Assuming that the particle is a solid uniform ball, and that its radius is small compared with the radius of curvature of the bowl $r_0$, the rotational energy due to the no-slipping rolling will be proportional to linear kinetic energy: $RKE = \frac 25 \frac {m v^2}{2}$. Then, your energy balance equation would be written: $$ KE_0+RKE_0 + PE_0 = KE_f+RKE_f. $$

Substituting the expression for rotational energy: $$ \frac 75 \frac{m v_0^2}{2} + m g h = \frac 75 \frac {m v_f^2} 2 $$ Note, that this equation could be made looking like your original energy conservation equation (without rolling) if we redefine parameters $m$ and $g$: $$ m' = \frac75 m, \qquad g' = \frac57 g, \qquad m g = m'g', $$ will gives us energy equation same as without rotation. Since mass can be eliminated from all equations, and energy equation is the only one containing $g$, such 'renormalization' is consistent with the rest of the equations.

Third point, is that you made a mistake. To understand that, take a limit $v_0 = 0$, $h\ne 0$, which correspond to particle sliding down the bowl without any angular momentum. Your answer will always give $\theta=\pi/2$, while the correct answer should be $\theta = \arcsin (h/r_0)$.

This mistake appeared in writing angular momentum conservation equation. You wrote: $$ mr_ov_o=mrv_f\cos \theta. $$ However, here $\theta$ is not the angle between the velocity and horizontal plane. The velocity of the particle has two components in the spherical coordinate system: $$ \mathbf{v}=\mathbf{e}_{\psi}\, v_\psi + \mathbf{e}_{\phi}\, v_\phi $$ where $\mathbf{e}_{\psi,\phi}$ are unit vectors in the directions of increasing nutation and azimuthal angles. $v_\phi$ enters the (correct) equation for angular momentum conservation: $$ m v_0 r_0 = m (v_f)_\phi r, $$ however, because the $\mathbf{e}_\psi$ vector is not vertical, $(v_f)_\phi \ne v_f \cos \theta $. Instead, since $\mathbf{e}_\psi$ forms an angle $\psi$ with vertical axis: $$ \sin \theta = \frac{(v_f)_\psi\, \cos \psi }{v_f} = \frac{(v_f)_\psi\, r }{v_f\, r_0}, $$ where we used $\cos \psi = r / r_0$. The unknown component $(v_f)_\psi$ could be easily found, since from the momentum conservation equation we know $(v_f)_\phi$, and from energy conservation (either your original or my for rolling ball) we can find $v_f$.

So after substitution we get $$ \sin \theta = \pm \sqrt{\frac{r^2}{r_0^2}-\frac{v_0^2}{v_f^2}}, $$ where $v_f$ is found using the energy equation: $v_f^2 = v_0^2 + 2 g \sqrt{r_0^2-r^2}$, (substitute $g$ with $5g/7$ to account for no-slip rolling). The $\pm$ sign reflects the fact that the velocity could have either positive or negative vertical component.

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If you assume that the particle is rolling without slippage, then the angular momentum moves between the rolling motion and the rotation in the bowl and your point 1 is not corrected. Regarding your point 3, $\cos(\theta)$ is fine if you consider the plane tangent to the sphere instead of the vertical one. –  DarioP Dec 6 '13 at 12:21
    
@DarioP: Wrong. If the particle is small, the additional angular momentum is negligible. For instance at the start, when velocity is $v_0$ and is horizontal: $L_{z\,\text{rot}}=I \omega = \frac25 m v_0 a \ll m v_0 r_0 $, where $a$ is radius of the particle. So as long as we neglect other $a/r_0$ corrections, there is no point to account for that one. –  user23660 Dec 6 '13 at 13:45
    
@DarioP: As for $\cos \theta$, the problem explicitly defines $\theta$ as an angle between velocity and horizontal. Measuring it in the in the inclined plane is wrong. –  user23660 Dec 6 '13 at 13:57
    
Well, I did not see the point in your answer where you were assuming $a/r_0 \ll 1$, yet the problem does not put any constraint on $v_0$ and $h$, what happens if $v_0 \rightarrow 0$ and $h\rightarrow r_0$? - If after "the horizontal" instead of "plane" you put "straight line tangent in B", then $\theta$ lies on an inclined plane and you don't need further complications. I wouldn't use the word "wrong", that's just a different, maybe a little more effective, point of view. –  DarioP Dec 6 '13 at 15:31
    
@DarioP: $a$ is defined by me in the comment above as the radius of the particle. The assumption $a/r \ll 1$ was already made implicitly in the formulation of a problem by OP (check for yourself where it was implied). It does not limit possible values of $v_0$ and $h$. As for the definition of $\theta$ take values $v_0=0$ and $h\ne 0$ and insert them into OP's answer. You will see that it is "wrong" (and not just (in)effective). –  user23660 Dec 6 '13 at 17:08

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