A particle \(P\) is projected horizontally with speed \(u \mathrm {~ms} ^ { - 1 }\) from the highest point of a smooth sphere of radius \(r \mathrm {~m}\) and centre \(O\). It moves on the surface in a vertical plane, and at a particular instant the radius \(O P\) makes an angle \(\theta\) with the upward vertical, as shown. At this instant \(P\) has speed \(v \mathrm {~ms} ^ { - 1 }\) and
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the magnitude of the reaction between \(P\) and the sphere is \(X \mathrm {~N}\).
Assuming that \(u ^ { 2 } < g r\), show that
\(v ^ { 2 } = u ^ { 2 } + 2 g r ( 1 - \cos \theta )\),
\(X = m g \left( 3 \cos \theta - 2 - \frac { y ^ { 2 } } { g r } \right)\).
(2 marks)
(4 marks)
Show that \(P\) leaves the surface of the sphere when \(\cos \theta = \frac { u ^ { 2 } + 2 g r } { 3 g r }\).
Discuss what happens if \(u ^ { 2 } \geq g r\).
A particle \(P\) of mass \(m \mathrm {~kg}\) hangs in equilibrium at one end of a light spring, of natural length \(l \mathrm {~m}\) and modulus of elasticity \(\lambda \mathrm { N }\), whose other end is fixed at a point vertically above \(P\). In this position the length of the spring is \(( l + e ) \mathrm { m }\). When \(P\) is displaced vertically through a small distance and released, it performs simple harmonic motion with 5 oscillations per second.
Show that \(\frac { \lambda } { l } = 100 \pi ^ { 2 } \mathrm {~m}\).
Express \(e\) in terms of \(g\).
Determine, in terms of \(m\) and \(l\), the magnitude of the tension in the spring when it is stretched to twice its natural length.
(a) Prove that the centre of mass of a uniform solid right circular cone of height \(h\) and base radius \(r\) is at a distance \(\frac { 3 h } { 4 }\) from the vertex.
An item of confectionery consists of a thin wafer in the form of a hollow right circular cone of height \(h\) and mass \(m\), filled with solid chocolate, also of mass \(m\), to a depth of \(k h\) as shown. The centre of mass of the item is at \(O\), the centre of the horizontal plane face
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of the chocolate.
Show that \(k = \frac { 8 h } { 15 }\). (3 marks)
In the packaging process, the cone has to move on a conveyor belt inclined at an angle \(\alpha\) to the horizontal as shown. If the belt is rough enough to prevent sliding, and the maximum value of \(\alpha\) for which
\includegraphics[max width=\textwidth, alt={}, center]{45b4918e-5d4c-470d-b725-e3b46900d190-2_284_445_2271_1560}
the cone does not topple is \(45 ^ { \circ }\),
find the radius of the base of the cone in terms of \(h\).