Force on pulley from string

5 \includegraphics[max width=\textwidth, alt={}, center]{b5873699-d207-4cad-9518-1321dc429c15-3_305_599_1717_774} Particles \(P\) and \(Q\) are attached to opposite ends of a light inextensible string. \(P\) is at rest on a rough horizontal table. The string passes over a small smooth pulley which is fixed at the edge of the table. \(Q\) hangs vertically below the pulley (see diagram). The force exerted on the string by the pulley has magnitude \(4 \sqrt { } 2 \mathrm {~N}\). The coefficient of friction between \(P\) and the table is 0.8 .

1. Show that the tension in the string is 4 N and state the mass of \(Q\).
2. Given that \(P\) is on the point of slipping, find its mass. A particle of mass 0.1 kg is now attached to \(Q\) and the system starts to move.
3. Find the tension in the string while the particles are in motion.

2 Particles \(A\) of mass 0.65 kg and \(B\) of mass 0.35 kg are attached to the ends of a light inextensible string which passes over a fixed smooth pulley. \(B\) is held at rest with the string taut and both of its straight parts vertical. The system is released from rest and the particles move vertically. Find the tension in the string and the magnitude of the resultant force exerted on the pulley by the string.

8. \begin{figure}[h] \end{figure} Two particles \(P\) and \(Q\) have masses 1.5 kg and 3 kg respectively. The particles are attached to the ends of a light inextensible string. Particle \(P\) is held at rest on a fixed rough horizontal table. The coefficient of friction between \(P\) and the table is \(\frac { 1 } { 5 }\). The string is parallel to the table and passes over a small smooth light pulley which is fixed at the edge of the table. Particle \(Q\) hangs freely at rest vertically below the pulley, as shown in Figure 3. Particle \(P\) is released from rest with the string taut and slides along the table. Assuming that \(P\) has not reached the pulley, find

1. the tension in the string during the motion,
2. the magnitude and direction of the resultant force exerted on the pulley by the string.

7. \begin{figure}[h] \end{figure} One end of a light inextensible string is attached to a particle \(A\) of mass \(5 m\). The other end of the string is attached to a particle \(B\) of mass \(3 m\). The string passes over a small, smooth, light fixed pulley. Particle \(A\) is held at rest with the string taut and the hanging parts of the string vertical, as shown in Figure 4. Particle A is released.

1. Find, in terms of \(m\) and \(g\), the magnitude of the force exerted on the pulley by the string while \(A\) is falling and before \(B\) hits the pulley.
2. State how, in your solution to part (a), you have used the fact that the pulley is smooth.

2. \begin{figure}[h] \end{figure} One end of a string is attached to a small ball \(P\) of mass \(4 m\).
The other end of the string is attached to another small ball \(Q\) of mass \(3 m\).
The string passes over a fixed pulley.
Ball \(P\) is held at rest with the string taut and the hanging parts of the string vertical, as shown in Figure 1. Ball \(P\) is released.
The string is modelled as being light and inextensible, the balls are modelled as particles, the pulley is modelled as being smooth and air resistance is ignored.

1. Using the model, find, in terms of \(m\) and \(g\), the magnitude of the force exerted on the pulley by the string while \(P\) is falling and before \(Q\) hits the pulley.
2. State one limitation of the model, apart from ignoring air resistance, that will affect the accuracy of your answer to part (a).

6. Corinne and her brother Dermot are lifted by their parents onto the two ends of a rope which is slung over a large, horizontal branch. When their parents let go of them Dermot, whose mass is 54 kg , begins to descend with an acceleration of \(1 \mathrm {~ms} ^ { - 2 }\). By modelling the children as a pair of particles connected by a light inextensible string, and the branch as a smooth pulley,

1. show that Corinne's mass is 44 kg ,
2. calculate the tension in the rope,
3. find the force on the branch. In a more sophisticated model, the branch is assumed to be rough.
4. Explain what effect this would have on the initial acceleration of the children.
   (1 mark)

7. \begin{figure}[h] \end{figure} Figure 3 shows a particle of mass 4 kg resting on the surface of a rough plane inclined at an angle of \(30 ^ { \circ }\) to the horizontal. It is connected by a light inextensible string passing over a smooth pulley at the top of the plane, to a particle of mass 5 kg which hangs freely. The coefficient of friction between the 4 kg mass and the plane is \(\mu\) and when the system is released from rest the 4 kg mass starts to move up the slope.

1. Show that the acceleration of the system is \(\frac { 1 } { 9 } ( 3 - 2 \mu \sqrt { 3 } ) \mathrm { g } \mathrm { ms } ^ { - 2 }\).
2. Hence, find the maximum value of \(\mu\). Given that \(\mu = \frac { 1 } { 2 }\),
3. find the tension in the string in terms of \(g\),
4. show that the magnitude of the force on the pulley is given by \(\frac { 5 } { 3 } ( 2 \sqrt { 3 } + 1 ) \mathrm { g }\). END

\includegraphics{figure_3} A particle \(A\) of mass 4 kg moves on the inclined face of a smooth wedge. This face is inclined at 30° to the horizontal. The wedge is fixed on horizontal ground. Particle \(A\) is connected to a particle \(B\), of mass 3 kg, by a light inextensible string. The string passes over a small light smooth pulley which is fixed at the top of the plane. The section of the string from \(A\) to the pulley lies in a line of greatest slope of the wedge. The particle \(B\) hangs freely below the pulley, as shown in Fig. 3. The system is released from rest with the string taut. For the motion before \(A\) reaches the pulley and before \(B\) hits the ground, find

1. the tension in the string, [6]
2. the magnitude of the resultant force exerted by the string on the pulley. [3]

1. The string in this question is described as being 'light'.
   1. Write down what you understand by this description.
   2. State how you have used the fact that the string is light in your answer to part (a). [2]

\includegraphics{figure_2} Two particles \(A\) and \(B\) have masses \(2m\) and \(3m\) respectively. The particles are attached to the ends of a light inextensible string. Particle \(A\) is held at rest on a smooth horizontal table. The string passes over a small smooth pulley which is fixed at the edge of the table. Particle \(B\) hangs at rest vertically below the pulley with the string taut, as shown in Figure 2. Particle \(A\) is released from rest. Assuming that \(A\) has not reached the pulley, find

1. the acceleration of \(B\), [5]
2. the tension in the string, [1]
3. the magnitude and direction of the force exerted on the pulley by the string. [4]

\includegraphics{figure_3} Figure 3 shows a particle \(X\) of mass 3 kg on a smooth plane inclined at an angle 30° to the horizontal, and a particle \(Y\) of mass 2 kg on a smooth plane inclined at an angle 60° to the horizontal. The two particles are connected by a light, inextensible string of length 2.5 metres passing over a smooth pulley at \(C\) which is the highest point of the two planes. Initially, \(Y\) is at a point just below \(C\) touching the pulley with the string taut. When the particles are released from rest they travel along the lines of greatest slope, \(AC\) in the case of \(X\) and \(BC\) in the case of \(Y\), of their respective planes. \(A\) and \(B\) are the points where the planes meet the horizontal ground and \(AB = 4\) metres.

1. Show that the initial acceleration of the system is given by \(\frac{g}{10}\left(2\sqrt{3} - 3\right)\) ms\(^{-2}\). [7 marks]
2. By finding the tension in the string, or otherwise, find the magnitude of the force exerted on the pulley and the angle that this force makes with the vertical. [7 marks]
3. Find, correct to 2 decimal places, the speed with which \(Y\) hits the ground. [4 marks]

\includegraphics{figure_2} Figure 2 shows a particle \(A\) of mass 5 kg, lying on a smooth horizontal table which is 0.9 m above the floor. A light inextensible string of length 0.7 m connects \(A\) to a particle \(B\) of mass 2 kg. The string passes over a smooth pulley which is fixed to the edge of the table and \(B\) hangs vertically 0.4 m below the pulley. When the system is released from rest,

1. show that the magnitude of the force exerted on the pulley is \(\frac{10\sqrt{5}}{7}\) g N. [7 marks]
2. find the speed with which \(A\) hits the pulley. [3 marks]

When \(A\) hits the pulley, the string breaks and \(B\) subsequently falls freely under gravity.

1. Find the speed with which \(B\) hits the ground. [4 marks]