\(\ \)\vspace{-1.5cm}
\noindent
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\par
A heavy smooth lamina of mass \(M\) is free to slide without rotation
along a straight line on a fixed smooth horizontal table. A smooth
groove \(ABC\) is inscribed in the lamina, as indicated in the above
diagram. The tangents to the groove at \(A\) and at \(B\) are parallel
to the line. When the lamina is stationary, a particle of mass \(m\)
(where \(m < M\)) enters the groove at \(A\). The particle is travelling,
with speed \(V\), parallel to the line and in the plane of the lamina
and table. Calculate the speeds of the particle and of the lamina,
when the particle leaves the groove at \(C\).
Suppose now that the lamina is held fixed by a peg attached to the
line. Supposing that the groove \(ABC\) is a semicircle of radius \(r\),
obtain the value of the average force per unit time exerted on the
peg by the lamina between the instant that the particle enters the
groove and the instant that it leaves it.