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    A marble of mass m is moving in a horizontal circle round the inside surface of a smooth hemispherical bowl of radius r. The centre of the circle is at a distance r/2 below the centre of the bowl. Find the magnitude of the reaction between the marble and the bowl and the speed of the marble.

    Any hints please? Having a massive mental block here..
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    I'll have a go.

    First of all, I tried to work out the radius of the circle that the marble is moving around. Looking at the bowl horizontally (and pretending that the bowl is a sphere rather than a hemisphere) gives the following diagram (with x being the required radius).



    Using trigonometry, we find

    \large{\cos\theta = \frac{(\frac{\mbox{r}}{2})}{\mbo  x{r}} = \frac{1}{2}}

    Therefore, \large{\theta = \cos^{-1} \frac{1}{2} = 60^o }.

    This thus gives \large{x = r\sin\theta = r\sin 60^o = \frac{r\sqrt 3}{2}}.

    Now, considering the forces acting on the marble, and once again looking at the bowl from the side, gives



    You know x and \large{\theta}, so you can resolve vertically and horizontally to answer your question (or you could just look at the spoiler).

    Spoiler:
    Show

    Resolving vertically gives:

    \large{\mbox{R}\cos\theta = mg}
    Therefore,
    Unparseable or potentially dangerous latex formula. Error 4: no dvi output from LaTeX. It is likely that your formula contains syntax errors or worse.
    \large{\mbox{R} = 2mg


    Resolving horizontally gives:

    \large{\mbox{R}\sin\theta = \frac{mv^2}{x}}
    Thus
    Unparseable or potentially dangerous latex formula. Error 4: no dvi output from LaTeX. It is likely that your formula contains syntax errors or worse.
    \large{v^2 = \frac{2\sqrt3 mgx}{2m} = xg\sqrt 3 = \frac{3rg}{2}

    And finally \large{v = \sqrt(\frac{3rg}{2})}
    Therefore, the marble's speed is \large{\sqrt(\frac{3rg}{2})}.


    Note that it's quite probable that I've made mistakes, but it should give you an idea as to how to do the question (I hope).

    EDIT: correcting grammar.
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Updated: June 12, 2006

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