SPS SPS FM Mechanics 2024 January — Question 5 13 marks

Exam BoardSPS
ModuleSPS FM Mechanics (SPS FM Mechanics)
Year2024
SessionJanuary
Marks13
TopicCircular Motion 2
TypeConical pendulum (horizontal circle)
DifficultyChallenging +1.2 This is a standard conical pendulum problem requiring resolution of forces in horizontal and vertical directions, with the added constraint of a cone surface providing a normal reaction. Part (i) involves straightforward force resolution and circular motion equations. Part (ii) requires eliminating the normal reaction using the inequality N>0 and manipulating the resulting expression—a typical Further Maths mechanics problem but more involved than basic circular motion questions.
Spec6.05b Circular motion: v=r*omega and a=v^2/r6.05c Horizontal circles: conical pendulum, banked tracks
5. A cone of semi-vertical angle \(60 ^ { \circ }\) is fixed with its axis vertical and vertex upwards. A particle of mass \(m\) is attached to one end of a light inextensible string of length \(l\). The other end of the string is attached to a fixed point vertically above the vertex of the cone. The particle moves in a horizontal circle on the smooth outer surface of the cone with constant angular speed \(\omega\), with the string making a constant angle \(60 ^ { \circ }\) with the horizontal, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{5f9a87c6-2255-4178-ab04-441bb0cc4ce0-10_538_648_456_664}
  1. Find the tension in the string, in terms of \(m , l , \omega\) and \(g\). The particle remains on the surface of the cone.
  2. Show that the time for the particle to make one complete revolution is greater than $$2 \pi \sqrt { \frac { l \sqrt { 3 } } { 2 g } } .$$ [Question 5 Continued]
    [0pt] [Question 5 Continued]
5.

A cone of semi-vertical angle $60 ^ { \circ }$ is fixed with its axis vertical and vertex upwards. A particle of mass $m$ is attached to one end of a light inextensible string of length $l$. The other end of the string is attached to a fixed point vertically above the vertex of the cone. The particle moves in a horizontal circle on the smooth outer surface of the cone with constant angular speed $\omega$, with the string making a constant angle $60 ^ { \circ }$ with the horizontal, as shown in the diagram.\\
\includegraphics[max width=\textwidth, alt={}, center]{5f9a87c6-2255-4178-ab04-441bb0cc4ce0-10_538_648_456_664}\\
(i) Find the tension in the string, in terms of $m , l , \omega$ and $g$.

The particle remains on the surface of the cone.\\
(ii) Show that the time for the particle to make one complete revolution is greater than

$$2 \pi \sqrt { \frac { l \sqrt { 3 } } { 2 g } } .$$

[Question 5 Continued]\\[0pt]
[Question 5 Continued]

\hfill \mbox{\textit{SPS SPS FM Mechanics 2024 Q5 [13]}}
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