Particle on rod or wire with friction

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\includegraphics[max width=\textwidth, alt={}, center]{01e73486-5a95-4e65-bf18-518d1adc7cfb-4_529_481_255_831} A small ring of mass 0.2 kg is threaded on a fixed vertical rod. The end \(A\) of a light inextensible string is attached to the ring. The other end \(C\) of the string is attached to a fixed point of the rod above \(A\). A horizontal force of magnitude 8 N is applied to the point \(B\) of the string, where \(A B = 1.5 \mathrm {~m}\) and \(B C = 2 \mathrm {~m}\). The system is in equilibrium with the string taut and \(A B\) at right angles to \(B C\) (see diagram).

1. Find the tension in the part \(A B\) of the string and the tension in the part \(B C\) of the string. The equilibrium is limiting with the ring on the point of sliding up the rod.
2. Find the coefficient of friction between the ring and the rod.

5. \begin{figure}[h] \end{figure} A small bead of mass 0.2 kg is attached to the end \(P\) of a light rod \(P Q\). The bead is threaded onto a fixed vertical rough wire. The bead is held in equilibrium with the \(\operatorname { rod } P Q\) inclined to the wire at an angle \(\alpha\), where \(\tan \alpha = \frac { 4 } { 3 }\), as shown in Figure 2. The thrust in the rod is \(T\) newtons.
The bead is modelled as a particle.

1. Find the magnitude and direction of the friction force acting on the bead when \(T = 2.5\) The coefficient of friction between the bead and the wire is \(\mu\).
   Given that the greatest possible value of \(T\) is 6.125
2. find the value of \(\mu\).

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\includegraphics[max width=\textwidth, alt={}, center]{99d30766-9c1b-43a8-986a-112b78b08146-3_697_579_1238_781} Two small rings \(A\) and \(B\) are attached to opposite ends of a light inextensible string. The rings are threaded on a rough wire which is fixed vertically. \(A\) is above \(B\). A horizontal force is applied to a point \(P\) of the string. Both parts \(A P\) and \(B P\) of the string are taut. The system is in equilibrium with angle \(B A P = \alpha\) and angle \(A B P = \beta\) (see diagram). The weight of \(A\) is 2 N and the tensions in the parts \(A P\) and \(B P\) of the string are 7 N and \(T \mathrm {~N}\) respectively. It is given that \(\cos \alpha = 0.28\) and \(\sin \alpha = 0.96\), and that \(A\) is in limiting equilibrium.

1. Find the coefficient of friction between the wire and the ring \(A\).
2. By considering the forces acting at \(P\), show that \(T \cos \beta = 1.96\).
3. Given that there is no frictional force acting on \(B\), find the mass of \(B\).

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\includegraphics[max width=\textwidth, alt={}, center]{4c8f0d10-ea1e-4aee-870d-71a52dd948ed-03_697_579_1238_781} Two small rings \(A\) and \(B\) are attached to opposite ends of a light inextensible string. The rings are threaded on a rough wire which is fixed vertically. \(A\) is above \(B\). A horizontal force is applied to a point \(P\) of the string. Both parts \(A P\) and \(B P\) of the string are taut. The system is in equilibrium with angle \(B A P = \alpha\) and angle \(A B P = \beta\) (see diagram). The weight of \(A\) is 2 N and the tensions in the parts \(A P\) and \(B P\) of the string are 7 N and \(T \mathrm {~N}\) respectively. It is given that \(\cos \alpha = 0.28\) and \(\sin \alpha = 0.96\), and that \(A\) is in limiting equilibrium.

1. Find the coefficient of friction between the wire and the ring \(A\).
2. By considering the forces acting at \(P\), show that \(T \cos \beta = 1.96\).
3. Given that there is no frictional force acting on \(B\), find the mass of \(B\). \section*{June 2005}