Standard +0.3 This is a standard M3 circular motion question with a particle on a sphere. Part (a) is routine circular motion formula application. Part (b) follows a well-established template: energy conservation to find v², resolving forces radially to show R in terms of θ, finding acceleration components, and determining when R=0 for leaving the surface. All techniques are standard for this topic with no novel insight required, making it slightly easier than average.
A moon of mass \(7.5 \times 10 ^ { 22 } \mathrm {~kg}\) moves round a planet in a circular path of radius \(3.8 \times 10 ^ { 8 } \mathrm {~m}\), completing one orbit in a time of \(2.4 \times 10 ^ { 6 } \mathrm {~s}\). Find the force acting on the moon.
Fig. 2 shows a fixed solid sphere with centre O and radius 4 m . Its surface is smooth. The point A on the surface of the sphere is 3.5 m vertically above the level of O . A particle P of mass 0.2 kg is placed on the surface at A and is released from rest. In the subsequent motion, when OP makes an angle \(\theta\) with the horizontal and P is still on the surface of the sphere, the speed of P is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the normal reaction acting on P is \(R \mathrm {~N}\).
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\begin{enumerate}[label=(\alph*)]
\item A moon of mass $7.5 \times 10 ^ { 22 } \mathrm {~kg}$ moves round a planet in a circular path of radius $3.8 \times 10 ^ { 8 } \mathrm {~m}$, completing one orbit in a time of $2.4 \times 10 ^ { 6 } \mathrm {~s}$. Find the force acting on the moon.
\item Fig. 2 shows a fixed solid sphere with centre O and radius 4 m . Its surface is smooth. The point A on the surface of the sphere is 3.5 m vertically above the level of O . A particle P of mass 0.2 kg is placed on the surface at A and is released from rest. In the subsequent motion, when OP makes an angle $\theta$ with the horizontal and P is still on the surface of the sphere, the speed of P is $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and the normal reaction acting on P is $R \mathrm {~N}$.
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\includegraphics[alt={},max width=\textwidth]{b7f8bdfd-33dc-4453-8f3a-ddd24be17372-3_746_734_705_662}
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\caption{Fig. 2}
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\begin{enumerate}[label=(\roman*)]
\item Express $v ^ { 2 }$ in terms of $\theta$.
\item Show that $R = 5.88 \sin \theta - 3.43$.
\item Find the radial and tangential components of the acceleration of P when $\theta = 40 ^ { \circ }$.
\item Find the value of $\theta$ at the instant when P leaves the surface of the sphere.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{OCR MEI M3 2006 Q2 [18]}}