Energy methods for pulley systems

2 \includegraphics[max width=\textwidth, alt={}, center]{ceb367ee-4e12-4cb2-9020-078ea5724d6e-2_529_691_529_726} Particle \(A\) of mass 1.6 kg and particle \(B\) of mass 2 kg are attached to opposite ends of a light inextensible string. The string passes over a small smooth pulley fixed at the top of a smooth plane, which is inclined at angle \(\theta\), where \(\sin \theta = 0.8\). Particle \(A\) is held at rest at the bottom of the plane and \(B\) hangs at a height of 3.24 m above the level of the bottom of the plane (see diagram). \(A\) is released from rest and the particles start to move.

1. Show that the loss of potential energy of the system, when \(B\) reaches the level of the bottom of the plane, is 23.328 J .
2. Hence find the speed of the particles when \(B\) reaches the level of the bottom of the plane.

4. \begin{figure}[h] \end{figure} Two particles \(A\) and \(B\), of mass \(m\) and \(2 m\) respectively, are attached to the ends of a light inextensible string. The particle \(A\) lies on a rough plane inclined at an angle \(\alpha\) to the horizontal, where tan \(\alpha = \frac { 3 } { 4 }\). The string passes over a small light smooth pulley \(P\) fixed at the top of the plane. The particle \(B\) hangs freely below \(P\), as shown in Figure 2. The particles are released from rest with the string taut and the section of the string from \(A\) to \(P\) parallel to a line of greatest slope of the plane. The coefficient of friction between \(A\) and the plane is \(\frac { 5 } { 8 }\). When each particle has moved a distance \(h , B\) has not reached the ground and \(A\) has not reached \(P\).

1. Find an expression for the potential energy lost by the system when each particle has moved a distance \(h\). When each particle has moved a distance \(h\), they are moving with speed \(v\). Using the workenergy principle,
2. find an expression for \(v ^ { 2 }\), giving your answer in the form \(k g h\), where \(k\) is a number.

2 Fig. 2 shows a wedge of angle \(30 ^ { \circ }\) fixed to a horizontal floor. Small objects P , of mass 8 kg , and Q , of mass 10 kg , are connected by a light inextensible string that passes over a smooth pulley at the top of the wedge. The part of the string between P and the pulley is parallel to a line of greatest slope of the wedge. Q hangs freely and at no time does either P or Q reach the pulley or P reach the floor. \begin{figure}[h] \end{figure}

1. Assuming the string remains taut, find the change in the gravitational potential energy of the system when Q descends \(h \mathrm {~m}\), stating whether it is a loss or a gain. Object P makes smooth contact with the wedge. The system is set in motion with the string taut.
2. Find the speed at which Q hits the floor if
   (A) the system is released from rest with Q a distance of 1.2 m above the floor,
   (B) instead, the system is set in motion with Q a distance of 0.3 m above the floor and P travelling down the slope at \(1.05 \mathrm {~ms} ^ { - 1 }\). The sloping face is roughened so that the coefficient of friction between object P and the wedge is 0.9 . The system is set in motion with the string taut and P travelling down the slope at \(2 \mathrm {~ms} ^ { - 1 }\).
3. How far does P move before it reaches its lowest point?
4. Determine what happens to the system after P reaches its lowest point.
5. Calculate the power of the frictional force acting on P in part (iii) at the moment the system is set in motion. \section*{Question 3 begins on page 4.}

7 Fig. 7.1 shows one end of a light inextensible string attached to a block A of mass 4.4 kg . The other end of the string is attached to a block B of mass 5.2 kg . Block A is in contact with a smooth horizontal plane. The string is taut and passes over a small smooth pulley at the end of the plane. Block B is inside a hollow vertical tube and the vertical sides of B are in contact with the tube. Initially B is 1.6 m above the horizontal base of the tube. \begin{figure}[h] \end{figure} The blocks are released from rest. It may be assumed that in the subsequent motion A does not reach the pulley and the string remains taut. Block B reaches the base of the tube with speed \(3.5 \mathrm {~ms} ^ { - 1 }\).

1. Given that the frictional force exerted by the tube on B is constant, use an energy method to show that the magnitude of this force is 14.21 N . Blocks A and B remain attached to the opposite ends of a light inextensible string, but A is now in contact with a rough plane inclined at \(\theta ^ { \circ }\) to the horizontal, as shown in Fig. 7.2. The string connecting A and B is taut and passes over a small smooth pulley at the top of the plane. Block B is inside the same hollow vertical tube as before with the vertical sides of B in contact with the tube. It may be assumed that the frictional force exerted by the tube on B remains unchanged. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{b20e2254-955e-466c-8161-9614d8ccdba0-8_623_723_552_260} \captionsetup{labelformat=empty} \caption{Fig. 7.2}
   \end{figure} The coefficient of friction between block A and the plane is \(\frac { 3 } { 11 }\).
   The blocks are released from rest, with block B 1.6 m above the base of the tube. It may be assumed that in the subsequent motion A does not reach the pulley and the string remains taut.
2. Given that block B reaches the base of the tube with speed \(0.7 \mathrm {~ms} ^ { - 1 }\), show that \(\theta\) satisfies the equation \(3 \cos \theta + 11 \sin \theta = k\),
   where \(k\) is a constant to be determined. \section*{END OF QUESTION PAPER} \section*{}

\includegraphics{figure_6_1} **Fig. 6.1** Fig. 6.1 shows particles \(A\) and \(B\), of masses 4 kg and 3 kg respectively, attached to the ends of a light inextensible string that passes over a small smooth pulley. The pulley is fixed at the top of a plane which is inclined at an angle of 30° to the horizontal. \(A\) hangs freely below the pulley and \(B\) is on the inclined plane. The string is taut and the section of the string between \(B\) and the pulley is parallel to a line of greatest slope of the plane.

1. It is given that the plane is rough and the particles are in limiting equilibrium. Find the coefficient of friction between \(B\) and the plane. [6]
2. \includegraphics{figure_6_2} **Fig. 6.2** It is given instead that the plane is smooth and the particles are released from rest when the difference in the vertical heights of the particles is 1 m (see Fig. 6.2). Use an energy method to find the speed of the particles at the instant when the particles are at the same horizontal level. [6]

Two particles \(P\) and \(Q\), of masses \(3\) kg and \(2\) kg respectively, rest on the smooth faces of a wedge whose cross-section is a triangle with angles \(30°\), \(60°\) and \(90°\), as shown. \(P\) and \(Q\) are connected by a light string, parallel to the lines of greatest slope of the two planes, which passes over a fixed pulley at the highest point of the wedge. \includegraphics{figure_6} The system is released from rest with \(P\) \(0.8\) m from the pulley and \(Q\) \(1\) m from the bottom of the wedge, and \(Q\) starts to move down. Calculate

1. the acceleration of either particle, \hfill [5 marks]
2. the tension in the string, \hfill [2 marks]
3. the speed with which \(P\) reaches the pulley. \hfill [3 marks]

Two modelling assumptions have been made about the string and the pulley.

1. State these two assumptions and briefly describe how you have used each one in your solution. \hfill [4 marks]

5 \includegraphics[max width=\textwidth, alt={}, center]{77976dad-c055-45fd-93fe-e37fa8e9ae22-3_343_691_254_725} A light inextensible rope has a block \(A\) of mass 5 kg attached at one end, and a block \(B\) of mass 16 kg attached at the other end. The rope passes over a smooth pulley which is fixed at the top of a rough plane inclined at an angle of \(30 ^ { \circ }\) to the horizontal. Block \(A\) is held at rest at the bottom of the plane and block \(B\) hangs below the pulley (see diagram). The coefficient of friction between \(A\) and the plane is \(\frac { 1 } { \sqrt { 3 } }\). Block \(A\) is released from rest and the system starts to move. When each of the blocks has moved a distance of \(x \mathrm {~m}\) each has speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Write down the gain in kinetic energy of the system in terms of \(v\).
2. Find, in terms of \(x\),
   (a) the loss of gravitational potential energy of the system,
   (b) the work done against the frictional force.
3. Show that \(21 v ^ { 2 } = 220 x\).