# OCR A Level: Mechanics 2 - Circular Motion [June 2010 Q5]

One end of a light inextensible string of length l is attached to the vertex of a smooth cone of semivertical angle $$45°$$. The cone is fixed to the ground with its axis vertical. The other end of the string is attached to a particle of mass $$m$$ which rotates in a horizontal circle in contact with the outer surface of the cone. The angular speed of the particle is $$ω$$ (see diagram). The tension in the string is $$T$$ and the contact force between the cone and the particle is $$R$$.

$$(\text{i})$$ By resolving horizontally and vertically, find two equations involving $$T$$ and $$R$$ and hence show that $$T=\dfrac{1}{2}m \left ( \sqrt{2} g + l \omega ^2 \right )$$.

$$(\text{ii})$$ When the string has length $$0.8 m$$, calculate the greatest value of $$ω$$ for which the particle remains in contact with the cone, to 3 significant figures.

Input $$100 \times$$ your answer to part $$(\text{ii})$$.

×