Two strings/rods system

4 One end of a light inextensible string of length 0.5 m is attached to a fixed point \(A\). The other end of the string is attached to a particle \(P\) of weight 6 N . Another light inextensible string of length 0.5 m connects \(P\) to a fixed point \(B\) which is 0.8 m vertically below \(A\). The particle \(P\) moves with constant speed in a horizontal circle with centre at the mid-point of \(A B\). Both strings are taut.

1. Calculate the speed of \(P\) when the tension in the string \(B P\) is 2 N .
2. Show that the angular speed of \(P\) must exceed \(5 \mathrm { rad } \mathrm { s } ^ { - 1 }\).

5 \includegraphics[max width=\textwidth, alt={}, center]{6503ebb1-5649-4ca5-9500-da4fb28009dd-3_540_537_255_804} A particle \(P\) of mass 0.2 kg is attached to a fixed point \(A\) by a light inextensible string of length 0.4 m . A second light inextensible string of length 0.3 m connects \(P\) to a fixed point \(B\) which is vertically below \(A\). The particle \(P\) moves in a horizontal circle, which has its centre on the line \(A B\), with the angle \(A P B = 90 ^ { \circ }\) (see diagram).

1. Given that the tensions in the two strings are equal, calculate the speed of \(P\).
2. It is given instead that \(P\) moves with its least possible angular speed for motion in this circle. Find this angular speed.

5. \begin{figure}[h] \end{figure} A particle \(P\) of mass \(m\) is attached to one end of a light inextensible string of length \(\mathrm { a } \sqrt { 3 }\). The other end of the string is attached to a fixed point A . The particle P is also attached to one end of a second light inextensible string of length a. The other end of this string is attached to a fixed point B , where B is vertically below A , with \(\mathrm { AB } = \mathrm { a }\). The particle \(P\) moves in a horizontal circle with centre 0 , where 0 is vertically below \(B\).
The particle P moves with constant angular speed \(\omega\), with both strings taut, as shown in Figure 5.

1. Show that the upper string makes an angle of \(30 ^ { \circ }\) with the downward vertical and the lower string makes an angle of \(60 ^ { \circ }\) with the downward vertical.
2. Show that the tension in the upper string is \(\frac { 1 } { 2 } m \sqrt { 3 } \left( 2 g - a \omega ^ { 2 } \right)\).
3. Show that \(\frac { 2 g } { 3 a } < \omega ^ { 2 } < \frac { 2 g } { a }\)
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4. \begin{figure}[h] \end{figure} A light inextensible string has its ends attached to two fixed points \(A\) and \(B\). The point \(A\) is vertically above \(B\) and \(A B = 7 a\). A particle \(P\) of mass \(m\) is fixed to the string and moves in a horizontal circle of radius \(3 a\) with angular speed \(\omega\). The centre of the circle is \(C\) where \(C\) lies on \(A B\) and \(A C = 4 a\), as shown in Figure 4. Both parts of the string are taut.

1. Show that the tension in \(A P\) is \(\frac { 5 } { 7 } m \left( 3 a \omega ^ { 2 } + g \right)\).
2. Find the tension in \(B P\).
3. Deduce that \(\omega \geqslant \frac { 1 } { 2 } \sqrt { } \left( \frac { g } { a } \right)\).

2

1. Fig. 2 shows a light inextensible string of length 3.3 m passing through a small smooth ring R of mass 0.27 kg . The ends of the string are attached to fixed points A and B , where A is vertically above \(B\). The ring \(R\) is moving with constant speed in a horizontal circle of radius \(1.2 \mathrm {~m} , \mathrm { AR } = 2.0 \mathrm {~m}\) and \(\mathrm { BR } = 1.3 \mathrm {~m}\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{b8573ee2-771c-4a93-88d9-346a9da94494-3_570_659_493_781} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure}
   1. Show that the tension in the string is 6.37 N .
   2. Find the speed of R .
2. One end of a light inextensible string of length 1.25 m is attached to a fixed point O . The other end is attached to a particle P of mass 0.2 kg . The particle P is moving in a vertical circle with centre O and radius 1.25 m , and when P is at the highest point of the circle there is no tension in the string.
   1. Show that when P is at the highest point its speed is \(3.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). For the instant when the string OP makes an angle of \(60 ^ { \circ }\) with the upward vertical, find
   2. the radial and tangential components of the acceleration of P ,
   3. the tension in the string.

2

1. The fixed point A is vertically above the fixed point B . A light inextensible string of length 5.4 m has one end attached to A and the other end attached to B. The string passes through a small smooth ring R of mass 0.24 kg , and R is moving at constant angular speed in a horizontal circle. The circle has radius 1.6 m , and \(\mathrm { AR } = 3.4 \mathrm {~m} , \mathrm { RB } = 2.0 \mathrm {~m}\), as shown in Fig. 2 . \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{5a0df44f-f8f0-44d4-b2f6-70a5314706f9-3_565_504_447_753} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure}
   1. Find the tension in the string.
   2. Find the angular speed of R .
2. A particle P of mass 0.3 kg is joined to a fixed point O by a light inextensible string of length 1.8 m . The particle P moves without resistance in part of a vertical circle with centre O and radius 1.8 m . When OP makes an angle of \(25 ^ { \circ }\) with the downward vertical, the tension in the string is 15 N .
   1. Find the speed of P when OP makes an angle of \(25 ^ { \circ }\) with the downward vertical.
   2. Find the tension in the string when OP makes an angle of \(60 ^ { \circ }\) with the upward vertical.
   3. Find the speed of P at the instant when the string becomes slack.