Circular motion with rod

A particle is attached to a light rod (not a string) that is hinged or fixed; the particle moves in a horizontal or vertical circle; find tension/thrust in rod or speed.

3 questions · Standard +0.8

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Edexcel M3 2021 January Q3
11 marks Standard +0.3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8a687d17-ec7e-463f-84dd-605f5c230db1-08_506_527_251_712} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A fairground ride consists of a cabin \(C\) that travels in a horizontal circle with a constant angular speed about a fixed vertical central axis. The cabin is attached to one end of each of two rigid arms, each of length 5 m . The other end of the top arm is attached to the fixed point \(A\) at the top of the central axis of the ride. The other end of the lower arm is attached to the fixed point \(B\) on the central axis, where \(A B\) is 8 m , as shown in Figure 2. Both arms are free to rotate about the central axis. The arms are modelled as light inextensible rods. The cabin, together with the people inside, is modelled as a particle. The cabin completes one revolution every 2 seconds. Given that the combined mass of the cabin and the people is 600 kg ,
  1. find
    1. the tension in the upper arm of the ride,
    2. the tension in the lower arm of the ride. In a refined model, it is assumed that both arms stretch to a length of 5.1 m .
  2. State how this would affect the sum of the tensions in the two arms, justifying your answer.
Edexcel M3 2021 June Q2
9 marks Standard +0.3
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b99b3eb0-9bca-42e3-bea9-3b0454a872db-04_374_1084_246_493} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a fairground ride that consists of a chair of mass \(m\) attached to one end of a rigid arm of length \(\frac { 5 a } { 4 }\). The other end of the arm is freely hinged to the rim of a thin horizontal circular disc of radius \(a\). The disc rotates with constant angular speed \(\omega\) about a vertical axis through the centre of the disc. As the ride rotates the arm remains in a vertical plane through the centre of the disc. The arm makes a constant angle \(\theta\) with the vertical, where \(\tan \theta = \frac { 3 } { 4 }\) The chair is modelled as a particle and the arm is modelled as a light rod.
  1. Find the tension in the arm in terms of \(m\) and \(g\)
  2. Find \(\omega\) in terms of \(a\) and \(g\)
Edexcel M5 Q5
8 marks Challenging +1.8
A uniform rod \(PQ\), of mass \(m\) and length \(2a\), is made to rotate in a vertical plane with constant angular speed \(\sqrt{\frac{g}{a}}\) about a fixed smooth horizontal axis through the end \(P\) of the rod. Show that, when the rod is inclined at an angle \(\theta\) to the downward vertical, the magnitude of the force exerted on the axis by the rod is \(2mg|\cos(\frac{1}{2}\theta)|\). [8]