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 \(5 \mathrm { rad } \mathrm { s } ^ { - 1 }\).

6
\includegraphics[max width=\textwidth, alt={}, center]{874622ab-4c75-4a32-bae5-eef780ed0cc0-10_757_464_258_836} A particle \(P\) of mass 0.2 kg is attached to one end of a light inextensible string of length 0.6 m . 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 0.6 m , the other end of which is attached to a fixed point \(B\) vertically below \(A\). The particle moves in a horizontal circle of radius 0.3 m , which has its centre at the mid-point of \(A B\), with both strings straight (see diagram).

1. Calculate the least possible angular speed of \(P\).
   The string \(A P\) will break if its tension exceeds 8 N . The string \(B P\) will break if its tension exceeds 5 N .
2. Find the greatest possible speed of \(P\).
   \(7 \quad\) A particle \(P\) of mass 0.2 kg is released from rest at a point \(O\) above horizontal ground. At time \(t \mathrm {~s}\) after its release the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) downwards. A vertically downwards force of magnitude \(0.6 t \mathrm {~N}\) acts on \(P\). A vertically upwards force of magnitude \(k \mathrm { e } ^ { - t } \mathrm {~N}\), where \(k\) is a constant, also acts on \(P\).
3. Show that \(\frac { \mathrm { d } v } { \mathrm {~d} t } = 10 - 5 k \mathrm { e } ^ { - t } + 3 t\).
4. Find the greatest value of \(k\) for which \(P\) does not initially move upwards.
5. Given that \(k = 1\), and that \(P\) strikes the ground when \(t = 2\), find the height of \(O\) above the ground. [5]
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

2 A particle \(P\) is projected with speed \(26 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(30 ^ { \circ }\) above the horizontal from a point \(O\) on a horizontal plane.

1. For the instant when the vertical component of the velocity of \(P\) is \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) downwards, find the direction of motion of \(P\) and the height of \(P\) above the plane.
2. \(P\) strikes the plane at the point \(A\). Calculate the time taken by \(P\) to travel from \(O\) to \(A\) and the distance \(O A\).
   \includegraphics[max width=\textwidth, alt={}, center]{022776e2-80bb-4e73-a309-8cd972ae7377-2_679_455_1544_845} Particles \(P\) and \(Q\) have masses 0.8 kg and 0.4 kg respectively. \(P\) is attached to a fixed point \(A\) by a light inextensible string which is inclined at an angle \(\alpha ^ { \circ }\) to the vertical. \(Q\) is attached to a fixed point \(B\), which is vertically below \(A\), by a light inextensible string of length 0.3 m . The string \(B Q\) is horizontal. \(P\) and \(Q\) are joined to each other by a light inextensible string which is vertical. The particles rotate in horizontal circles of radius 0.3 m about the axis through \(A\) and \(B\) with constant angular speed \(5 \mathrm { rad } \mathrm { s } ^ { - 1 }\) (see diagram).
3. By considering the motion of \(Q\), find the tensions in the strings \(P Q\) and \(B Q\).
4. Find the tension in the string \(A P\) and the value of \(\alpha\).

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 \(\frac { 2 g } { 3 a } < \omega ^ { 2 } < \frac { 2 g } { a }\)
   \(\_\_\_\_\) VIAV SIHI NI JIIHM ION OC
   VILU SIHIL NI GLIUM ION OC
   VEYV SIHI NI III HM ION OC

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 \(\omega \geqslant \frac { 1 } { 2 } \sqrt { } \left( \frac { g } { a } \right)\).

3 Two uniform small smooth spheres \(A\) and \(B\) have equal radii and each has mass \(m\). Sphere \(A\) is moving with speed \(u\) on a smooth horizontal surface when it collides directly with sphere \(B\) which is at rest. The coefficient of restitution between the spheres is \(\frac { 2 } { 3 }\). Sphere \(B\) is initially at a distance \(d\) from a fixed smooth vertical wall which is perpendicular to the direction of motion of \(A\). The coefficient of restitution between \(B\) and the wall is \(\frac { 1 } { 3 }\).

1. Show that the speed of \(B\) after its collision with the wall is \(\frac { 5 } { 18 } u\).
2. Find the distance of \(B\) from the wall when it collides with \(A\) for the second time.
   \includegraphics[max width=\textwidth, alt={}, center]{b10d2991-abff-4d2b-b470-1df844d1c7ee-08_743_673_258_737} A uniform rod \(A B\) of length \(3 a\) and weight \(W\) is freely hinged to a fixed point at the end \(A\). The end \(B\) is below the level of \(A\) and is attached to one end of a light elastic string of natural length 4a. The other end of the string is attached to a point \(O\) on a vertical wall. The horizontal distance between \(A\) and the wall is \(5 a\). The string and the rod make angles \(\theta\) and \(2 \theta\) respectively with the horizontal (see diagram). The system is in equilibrium with the rod and the string in the same vertical plane. It is given that \(\sin \theta = \frac { 3 } { 5 }\) and you may use the fact that \(\cos 2 \theta = \frac { 7 } { 25 }\).

7. A particle \(P\) of mass \(m \mathrm {~kg}\) is attached to points \(C\) and \(D\) on the same horizontal level by means of two light inextensible strings \(C P\) and \(D P\), both of length \(40 \mathrm {~cm} . P\) is projected with speed \(u \mathrm {~ms} ^ { - 1 }\) so as to move in a vertical circle in a plane perpendicular to \(C D\), so that angle \(P C D =\) angle \(P D C = \theta\) throughout the motion.
\includegraphics[max width=\textwidth, alt={}, center]{627b3411-07ba-4ee4-a672-93a64eeb90b3-2_335_405_1775_1572} If \(u\) is just large enough for the strings to remain taut as \(P\) describes this circular path,

1. show that \(u ^ { 2 } = 2 g \sin \theta\). The string \(D P\) breaks when \(P\) is at its lowest point. \(P\) then immediately starts to move in a horizontal circle on the end of the string \(C P\).
2. Prove that \(\tan \theta = \frac { 1 } { 5 } \sqrt { 5 }\).

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.