Standard +0.3 This is a standard circular motion problem on a curved surface requiring resolution of forces (normal reaction and weight) and application of F=ma towards the centre. The geometry is straightforward (finding the angle from the given vertical distance), and the method is a well-practiced technique in Further Maths mechanics. It's slightly easier than average due to the clear setup and standard approach, though it requires careful geometric reasoning and algebraic manipulation across 6 marks.
\includegraphics{figure_3}
A hemispherical shell of radius \(a\) is fixed with its rim uppermost and horizontal. A small bead, \(B\), is moving with constant angular speed, \(\omega\), in a horizontal circle on the smooth inner surface of the shell. The centre of the path of \(B\) is at a distance \(\frac{1}{4}a\) vertically below the level of the rim of the hemisphere, as shown in Figure 1.
Find the magnitude of \(\omega\), giving your answer in terms of \(a\) and \(g\).
[6]
\includegraphics{figure_3}
A hemispherical shell of radius $a$ is fixed with its rim uppermost and horizontal. A small bead, $B$, is moving with constant angular speed, $\omega$, in a horizontal circle on the smooth inner surface of the shell. The centre of the path of $B$ is at a distance $\frac{1}{4}a$ vertically below the level of the rim of the hemisphere, as shown in Figure 1.
Find the magnitude of $\omega$, giving your answer in terms of $a$ and $g$.
[6]
\hfill \mbox{\textit{SPS SPS FM Mechanics 2021 Q5 [6]}}