Connected particles pulley energy method

3. Two particles \(P\) and \(Q\), of mass \(2 m\) and \(3 m\) respectively, are connected by a light inextensible string. Initially \(P\) is held at rest on a fixed rough plane inclined at \(\theta\) to the horizontal ground, where \(\sin \theta = \frac { 2 } { 5 }\). The string passes over a small smooth pulley fixed at the top of the plane. The particle \(Q\) hangs freely below the pulley, as shown in Figure 1. The part of the string from \(P\) to the pulley lies along a line of greatest slope of the plane. At time \(t = 0\) the system is released from rest with the string taut. When \(P\) moves the friction between \(P\) and the plane is modelled as a constant force of magnitude \(\frac { 3 } { 5 } m g\). At the instant when each particle has moved a distance \(d\), they are both moving with speed \(v\), particle \(P\) has not reached the pulley and \(Q\) has not reached the ground.

1. Show that the total potential energy lost by the system when each particle has moved a distance \(d\) is \(\frac { 11 } { 5 } m g d\).
2. Use the work-energy principle to find \(v ^ { 2 }\) in terms of \(g\) and \(d\). When \(t = T\) seconds, \(d = 1.5 \mathrm {~m}\).
3. Find the value of \(T\).
   DO NOT WIRITE IN THIS AREA

6. \begin{figure}[h] \end{figure} Two particles, \(A\) and \(B\), of mass 2 kg and 3 kg respectively, are connected by a light inextensible string. Particle \(A\) is held at rest at the point \(X\) on a fixed rough ramp that is inclined at an angle \(\theta\) to the horizontal, where \(\tan \theta = \frac { 5 } { 12 }\). The string passes over a small smooth pulley \(P\) that is fixed at the top of the ramp. Particle \(B\) hangs vertically below \(P\), 2 m above the ground, as shown in Figure 4. The particles are released from rest with the string taut so that \(A\) moves up the ramp and the section of the string from \(A\) to \(P\) is parallel to a line of greatest slope of the ramp. The coefficient of friction between \(A\) and the ramp is \(\frac { 3 } { 8 }\) Air resistance is ignored.

1. Find the potential energy lost by the system as \(A\) moves 2 m up the ramp.
2. Find the work done against friction as \(A\) moves 2 m up the ramp. When \(B\) hits the ground, \(B\) is brought to rest by the impact and does not rebound and \(A\) continues to move up the ramp.
3. Use the work-energy principle to find the speed of \(B\) at the instant before it hits the ground. Particle \(A\) comes to instantaneous rest at the point \(Y\) on the ramp, where \(X Y = ( 2 + d ) \mathrm { m }\).
4. Use the work-energy principle to find the value of \(d\).

4. \begin{figure}[h] \end{figure} Two particles \(P\) and \(Q\), of mass 2 kg and 4 kg respectively, are connected by a light inextensible string. Initially \(P\) is held at rest at the point \(A\) on a rough fixed plane inclined at \(\alpha\) to the horizontal ground, where \(\sin \alpha = \frac { 3 } { 5 }\). The string passes over a small smooth pulley fixed at the top of the plane. The particle \(Q\) hangs freely below the pulley and 2.5 m above the ground, as shown in Figure 1. The part of the string from \(P\) to the pulley lies along a line of greatest slope of the plane. The system is released from rest with the string taut. At the instant when \(Q\) hits the ground, \(P\) is at the point \(B\) on the plane. The coefficient of friction between \(P\) and the plane is \(\frac { 1 } { 4 }\).

1. Find the work done against friction as \(P\) moves from \(A\) to \(B\).
2. Find the total potential energy lost by the system as \(P\) moves from \(A\) to \(B\).
3. Find, using the work-energy principle, the speed of \(P\) as it passes through \(B\).

4. \begin{figure}[h] \end{figure} Two particles \(P\) and \(Q\), of mass 2 kg and 4 kg respectively, are connected by a light inextensible string. Initially \(P\) is held at rest at the point \(A\) on a rough fixed plane inclined at \(\alpha\) to the horizontal ground, where \(\sin \alpha = \frac { 3 } { 5 }\). The string passes over a small smooth pulley fixed at the top of the plane. The particle \(Q\) hangs freely below the pulley and 2.5 m above the ground, as shown in Figure 1. The part of the string from \(P\) to the pulley lies along a line of greatest slope of the plane. The system is released from rest with the string taut. At the instant when \(Q\) hits the ground, \(P\) is at the point \(B\) on the plane. The coefficient of friction between \(P\) and the plane is \(\frac { 1 } { 4 }\).

1. Find the work done against friction as \(P\) moves from \(A\) to \(B\).
2. Find the total potential energy lost by the system as \(P\) moves from \(A\) to \(B\).
3. Find, using the work-energy principle, the speed of \(P\) as it passes through \(B\).

4 A block of mass 20 kg is pulled by a light, horizontal string over a rough, horizontal plane. During 6 seconds, the work done against resistances is 510 J and the speed of the block increases from \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Calculate the power of the pulling force. The block is now put on a rough plane that is at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 3 } { 5 }\). The frictional resistance to sliding is \(11 g \mathrm {~N}\). A light string parallel to the plane is connected to the block. The string passes over a smooth pulley and is connected to a freely hanging sphere of mass \(m \mathrm {~kg}\), as shown in Fig. 4. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{c1785fde-a6ce-4f8b-9948-4b4dd973ce84-6_348_855_847_605} \captionsetup{labelformat=empty} \caption{Fig. 4}
   \end{figure} In parts (ii) and (iii), the sphere is pulled downwards and then released when travelling at a speed of \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) vertically downwards. The block never reaches the pulley.
2. Suppose that \(m = 5\) and that after the sphere is released the block moves \(x \mathrm {~m}\) up the plane before coming to rest.
   (A) Find an expression in terms of \(x\) for the change in gravitational potential energy of the system, stating whether this is a gain or a loss.
   (B) Find an expression in terms of \(x\) for the work done against friction.
   (C) Making use of your answers to parts (A) and (B), find the value of \(x\).
3. Suppose instead that \(m = 15\). Calculate the speed of the sphere when it has fallen a distance 0.5 m from its point of release.

6. \begin{figure}[h] \end{figure} Two blocks, \(A\) and \(B\), of masses 2 kg and 4 kg respectively are attached to the ends of a light inextensible string. Initially \(A\) is held on a fixed rough plane. The plane is inclined to horizontal ground at an angle \(\theta\), where \(\tan \theta = \frac { 3 } { 4 }\) The string passes over a small smooth light pulley \(P\) that is fixed at the top of the plane. The part of the string from \(A\) to \(P\) is parallel to a line of greatest slope of the plane. Block \(A\) is held on the plane with the distance \(A P\) greater than 3 m .
Block \(B\) hangs freely below \(P\) at a distance of 3 m above the ground, as shown in Figure 4. The coefficient of friction between \(A\) and the plane is \(\mu\) Block \(A\) is released from rest with the string taut.
By modelling the blocks as particles,

1. find the potential energy lost by the whole system as a result of \(B\) falling 3 m . Given that the speed of \(B\) at the instant it hits the ground is \(4.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and ignoring air resistance,
2. use the work-energy principle to find the value of \(\mu\) After \(B\) hits the ground, \(A\) continues to move up the plane but does not reach the pulley in the subsequent motion.
   Block \(A\) comes to instantaneous rest after moving a total distance of ( \(3 + d\) ) m from its point of release. Ignoring air resistance,
3. use the work-energy principle to find the value of \(d\) \includegraphics[max width=\textwidth, alt={}, center]{86a37170-046f-46e5-9c8c-06d5f98ca4fe-20_2255_50_309_1981}