![(GBR 3)](/media/m/3/0/8/308bc67d981b1e70f413418d3fa4d9c9.png)
A smooth solid consists of a right circular cylinder of height
![h](/media/m/e/4/3/e438ac862510e579cf5cbdbe5904d4ba.png)
and base-radius
![r](/media/m/3/d/f/3df7cc5bbfb7b3948b16db0d40571068.png)
, surmounted by a hemisphere of radius
![r](/media/m/3/d/f/3df7cc5bbfb7b3948b16db0d40571068.png)
and center
![O.](/media/m/8/c/f/8cfaee47aa222d4f9d799d2e79461ae5.png)
The solid stands on a horizontal table. One end of a string is attached to a point on the base. The string is stretched (initially being kept in the vertical plane) over the highest point of the solid and held down at the point
![P](/media/m/9/6/8/968d210d037e7e95372de185e8fb8759.png)
on the hemisphere such that
![OP](/media/m/3/d/8/3d8977136b7195b09c5a409d5124d8cb.png)
makes an angle
![\alpha](/media/m/f/c/3/fc35d340e96ae7906bf381cae06e4d59.png)
with the horizontal. Show that if
![\alpha](/media/m/f/c/3/fc35d340e96ae7906bf381cae06e4d59.png)
is small enough, the string will slacken if slightly displaced and no longer remain in a vertical plane. If then pulled tight through
![P](/media/m/9/6/8/968d210d037e7e95372de185e8fb8759.png)
, show that it will cross the common circular section of the hemisphere and cylinder at a point
![Q](/media/m/4/5/c/45ce8d14aa1eb54f755fd8e332280abd.png)
such that
![\angle SOQ = \phi](/media/m/f/2/a/f2a7291667a5a16f292eca76ace9f870.png)
,
![S](/media/m/c/6/3/c63593c3ec0773fa38c2659e08119a75.png)
being where it initially crossed this section, and
![\sin \phi = \frac{r \tan \alpha}{h}](/media/m/6/8/4/684d3c06c75feb9d57c8b5ecdccf9598.png)
.
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$(GBR 3)$ A smooth solid consists of a right circular cylinder of height $h$ and base-radius $r$, surmounted by a hemisphere of radius $r$ and center $O.$ The solid stands on a horizontal table. One end of a string is attached to a point on the base. The string is stretched (initially being kept in the vertical plane) over the highest point of the solid and held down at the point $P$ on the hemisphere such that $OP$ makes an angle $\alpha$ with the horizontal. Show that if $\alpha$ is small enough, the string will slacken if slightly displaced and no longer remain in a vertical plane. If then pulled tight through $P$, show that it will cross the common circular section of the hemisphere and cylinder at a point $Q$ such that $\angle SOQ = \phi$, $S$ being where it initially crossed this section, and $\sin \phi = \frac{r \tan \alpha}{h}$.