Ring on wire with string

3 \includegraphics[max width=\textwidth, alt={}, center]{4a2bad7c-6720-414c-b336-060afb2255e9-05_610_591_257_778} A ring of mass 4 kg is threaded on a smooth circular rigid wire with centre \(C\). The wire is fixed in a vertical plane and the ring is kept at rest by a light string connected to \(A\), the highest point of the circle. The string makes an angle of \(25 ^ { \circ }\) to the vertical (see diagram). Find the tension in the string and the magnitude of the normal reaction of the wire on the ring.

6 \includegraphics[max width=\textwidth, alt={}, center]{d3bb6702-231d-42a0-830e-9f844dca78d7-3_387_1095_1724_525} A small smooth ring \(R\), of mass 0.6 kg , is threaded on a light inextensible string of length 100 cm . One end of the string is attached to a fixed point \(A\). A small bead \(B\) of mass 0.4 kg is attached to the other end of the string, and is threaded on a fixed rough horizontal rod which passes through \(A\). The system is in equilibrium with \(B\) at a distance of 80 cm from \(A\) (see diagram).

1. Find the tension in the string.
2. Find the frictional and normal components of the contact force acting on \(B\).
3. Given that the equilibrium is limiting, find the coefficient of friction between the bead and the rod.

7 \includegraphics[max width=\textwidth, alt={}, center]{d5f48bef-2518-4abd-b3e1-5e48ce56cf62-4_657_618_255_760} A small ring \(R\) is attached to one end of a light inextensible string of length 70 cm . A fixed rough vertical wire passes through the ring. The other end of the string is attached to a point \(A\) on the wire, vertically above \(R\). A horizontal force of magnitude 5.6 N is applied to the point \(J\) of the string 30 cm from \(A\) and 40 cm from \(R\). The system is in equilibrium with each of the parts \(A J\) and \(J R\) of the string taut and angle \(A J R\) equal to \(90 ^ { \circ }\) (see diagram).

1. Find the tension in the part \(A J\) of the string, and find the tension in the part \(J R\) of the string. The ring \(R\) has mass 0.2 kg and is in limiting equilibrium, on the point of moving up the wire.
2. Show that the coefficient of friction between \(R\) and the wire is 0.341 , correct to 3 significant figures. A particle of mass \(m \mathrm {~kg}\) is attached to \(R\) and \(R\) is now in limiting equilibrium, on the point of moving down the wire.
3. Given that the coefficient of friction is unchanged, find the value of \(m\). {www.cie.org.uk} after the live examination series. }

6 \includegraphics[max width=\textwidth, alt={}, center]{5cba3e17-3979-4c22-a415-2cdd60f09289-3_579_469_1142_840} One end of a light inextensible string is attached to a fixed point \(A\) of a fixed vertical wire. The other end of the string is attached to a small ring \(B\), of mass 0.2 kg , through which the wire passes. A horizontal force of magnitude 5 N is applied to the mid-point \(M\) of the string. The system is in equilibrium with the string taut, with \(B\) below \(A\), and with angles \(A B M\) and \(B A M\) equal to \(30 ^ { \circ }\) (see diagram).

1. Show that the tension in \(B M\) is 5 N .
2. The ring is on the point of sliding up the wire. Find the coefficient of friction between the ring and the wire.
3. A particle of mass \(m \mathrm {~kg}\) is attached to the ring. The ring is now on the point of sliding down the wire. Given that the coefficient of friction between the ring and the wire is unchanged, find the value of \(m\).

4 \includegraphics[max width=\textwidth, alt={}, center]{a9f3480e-7a8a-497d-a26a-b2aba9b05512-3_335_751_264_696} A particle \(P\) of weight 5 N is attached to one end of each of two light inextensible strings of lengths 30 cm and 40 cm . The other end of the shorter string is attached to a fixed point \(A\) of a rough rod which is fixed horizontally. A small ring \(S\) of weight \(W \mathrm {~N}\) is attached to the other end of the longer string and is threaded on to the rod. The system is in equilibrium with the strings taut and \(A S = 50 \mathrm {~cm}\) (see diagram).

1. By resolving the forces acting on \(P\) in the direction of \(P S\), or otherwise, find the tension in the longer string.
2. Find the magnitude of the frictional force acting on \(S\).
3. Given that the coefficient of friction between \(S\) and the rod is 0.75 , and that \(S\) is in limiting equilibrium, find the value of \(W\).

A uniform rod \(AB\), of length \(2a\) and mass \(2m\), can rotate freely in a vertical plane about a smooth horizontal axis through \(A\). A small rough ring of mass \(m\) is threaded on the rod. The rod is held in a horizontal position with the ring at rest at the mid-point of the rod. The rod is released from rest. Using energy considerations, show that, until the ring slides, $$a\dot{\theta}^2 = \frac{18}{11}g \sin \theta,$$ where \(\theta\) is the angle turned through by the rod. [3] Show that, until the ring slides, the magnitudes of the friction force and normal contact force acting on the ring are \(\frac{20}{11}mg \sin \theta\) and \(\frac{2}{11}mg \cos \theta\) respectively. [6] The coefficient of friction between the ring and the rod is \(\mu\). Find, in terms of \(\mu\), the value of \(\theta\) when the ring starts to slide. [2]

A ring of weight \(W\), with radius \(a\) and centre \(O\), is at rest on a rough surface that is inclined to the horizontal at an angle \(\alpha\) where \(\tan\alpha = \frac{1}{3}\). The plane of the ring is perpendicular to the inclined surface and parallel to a line of greatest slope of the surface. The point \(P\) on the circumference of the ring is such that \(OP\) is parallel to the surface. A light inextensible string is attached to \(P\) and to the point \(Q\), which is on the surface, such that \(PQ\) is horizontal (see diagram). The points \(O\), \(P\) and \(Q\) are in the same vertical plane. The system is in limiting equilibrium and the coefficient of friction between the ring and the surface is \(\mu\). \includegraphics{figure_4}

1. Find, in terms of \(W\), the tension in the string \(PQ\). [4]
2. Find the value of \(\mu\). [3]

\includegraphics{figure_4} A ring of weight \(W\), with radius \(a\) and centre \(O\), is at rest on a rough surface that is inclined to the horizontal at an angle \(\alpha\) where \(\tan\alpha = \frac{1}{3}\). The plane of the ring is perpendicular to the inclined surface and parallel to a line of greatest slope of the surface. The point \(P\) on the circumference of the ring is such that \(OP\) is parallel to the surface. A light inextensible string is attached to \(P\) and to the point \(Q\), which is on the surface, such that \(PQ\) is horizontal (see diagram). The points \(O\), \(P\) and \(Q\) are in the same vertical plane. The system is in limiting equilibrium and the coefficient of friction between the ring and the surface is \(\mu\).

1. Find, in terms of \(W\), the tension in the string \(PQ\). [4]
2. Find the value of \(\mu\). [3]

\includegraphics{figure_3} A uniform square lamina of side \(2a\) and weight \(W\) is suspended from a light inextensible string attached to the midpoint \(E\) of the side \(AB\). The other end of the string is attached to a fixed point \(P\) on a rough vertical wall. The vertex \(B\) of the lamina is in contact with the wall. The string \(EP\) is perpendicular to the side \(AB\) and makes an angle \(\theta\) with the wall (see diagram). The string and the lamina are in a vertical plane perpendicular to the wall. The coefficient of friction between the wall and the lamina is \(\frac{1}{2}\). Given that the vertex \(B\) is about to slip up the wall, find the value of \(\tan\theta\). [8]