Standard +0.3 This is a standard circular motion problem requiring resolution of forces (normal reaction and weight) and application of centripetal force formula. The setup is familiar from textbooks, the speed is given directly, and it requires straightforward geometric reasoning (relating height x to radius of circular path) followed by algebraic manipulation. Slightly easier than average due to the clean given speed and standard method.
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 \(\sqrt{\frac{g}{2}a}\).
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 $\sqrt{\frac{g}{2}a}$.
Find $x$ in terms of $a$. [6]
\hfill \mbox{\textit{SPS SPS FM Mechanics 2026 Q4 [6]}}