\includegraphics{figure_3} A particle moves on the inner surface of a smooth hollow cone of semi-vertical angle \(\alpha\). The axis of the cone is vertical with the vertex at the bottom. The particle moves in a horizontal circle of radius \(r\) with constant speed \(v\). Find expressions for the normal reactions on the particle from the cone surface, and show that the height of the particle above the vertex is \(\frac{v^2}{g \tan \alpha}\). [8]

Question 3:
3 | θ
Rcos = 0.5 g (=5)
Rsin θ = 0.5 × 5 2 × 0.4 (= 5)
tan θ =(0.5 × 5 2 × 0.4)/(0.5 g)
θ o
= 45 AG
R = 0.5 g/cos45
R = 7.07 N | B1
M1
M1
A1
M1
A1 | [6] | Resolving vertically
Use of N2L horizontally with
n w2r
acc =
Eliminating R
R 2 = (0.5 × 5 2 × 0.4) 2 + (0.5 g) 2
7.071.. | 6

\includegraphics{figure_3}

A particle moves on the inner surface of a smooth hollow cone of semi-vertical angle $\alpha$. The axis of the cone is vertical with the vertex at the bottom. The particle moves in a horizontal circle of radius $r$ with constant speed $v$.

Find expressions for the normal reactions on the particle from the cone surface, and show that the height of the particle above the vertex is $\frac{v^2}{g \tan \alpha}$.
[8]

\hfill \mbox{\textit{CAIE M2 2013 Q3 [8]}}