Standard +0.8 This is a standard circular motion problem requiring resolution of forces (normal reaction and weight) and application of centripetal force formula, but involves careful geometric reasoning to relate radius of circular path to height x, and algebraic manipulation to solve for x. The multi-step nature and need to correctly set up the geometry elevates it above average difficulty, though it follows a well-established method for conical pendulum/bowl problems.
A hollow hemispherical bowl of radius \(a\) has a smooth inner surface and is fixed with its axis vertical. A particle \(P\) of mass \(m\) moves in horizontal circles on the inner surface of the bowl, at a height \(x\) above the lowest point of the bowl. The speed of \(P\) is \(\sqrt{\frac{8}{3}ga}\).
Find \(x\) in terms of \(a\). [6]
A hollow hemispherical bowl of radius $a$ has a smooth inner surface and is fixed with its axis vertical. A particle $P$ of mass $m$ moves in horizontal circles on the inner surface of the bowl, at a height $x$ above the lowest point of the bowl. The speed of $P$ is $\sqrt{\frac{8}{3}ga}$.
Find $x$ in terms of $a$. [6]
\hfill \mbox{\textit{CAIE Further Paper 3 2021 Q2 [6]}}