Ring on vertical rod equilibrium

2 \includegraphics[max width=\textwidth, alt={}, center]{d0fa61eb-f320-427e-8883-de224d293933-2_701_323_1244_913} A small ring of mass 0.6 kg is threaded on a rough rod which is fixed vertically. The ring is in equilibrium, acted on by a force of magnitude 5 N pulling upwards at \(30 ^ { \circ }\) to the vertical (see diagram).

1. Show that the frictional force acting on the ring has magnitude 1.67 N , correct to 3 significant figures.
2. The ring is on the point of sliding down the rod. Find the coefficient of friction between the ring and the rod.

6 \includegraphics[max width=\textwidth, alt={}, center]{155bc571-80e4-4c93-859f-bb150a109211-3_465_410_1891_865} The diagram shows a ring of mass 2 kg threaded on a fixed rough vertical rod. A light string is attached to the ring and is pulled upwards at an angle of \(30 ^ { \circ }\) to the horizontal. The tension in the string is \(T \mathrm {~N}\). The coefficient of friction between the ring and the rod is 0.24 . Find the two values of \(T\) for which the ring is in limiting equilibrium.

5. \begin{figure}[h] \end{figure} A small ring of mass 0.2 kg is attached to one end of a light inextensible string.
The ring is threaded onto a fixed rough vertical rod.
The string is taut and makes an angle \(\theta\) with the rod, as shown in Figure 3, where \(\tan \theta = \frac { 12 } { 5 }\) Given that the ring is in equilibrium and that the tension in the string is 10 N ,

1. find the magnitude of the frictional force acting on the ring,
2. state the direction of the frictional force acting on the ring. The coefficient of friction between the ring and the rod is \(\frac { 1 } { 4 }\) Given that the ring is in equilibrium, and that the tension in the string, \(T\) newtons, can now vary,
3. 1. find the minimum value of \(T\)
   2. find the maximum value of \(T\)

\includegraphics{figure_1} A small ring \(P\) of mass \(0.03\) kg is threaded on a rough vertical rod. A light inextensible string is attached to the ring and is pulled upwards at an angle of \(15°\) to the horizontal. The tension in the string is \(2.5\) N (see diagram). The ring is in limiting equilibrium and on the point of sliding up the rod. Find the coefficient of friction between the ring and the rod. [4]

\includegraphics{figure_5} A ring of mass 4 kg is threaded on a fixed rough vertical rod. A light string is attached to the ring, and is pulled with a force of magnitude \(T\) N acting at an angle of \(60°\) to the downward vertical (see diagram). The ring is in equilibrium.

1. The normal and frictional components of the contact force exerted on the ring by the rod are \(R\) N and \(F\) N respectively. Find \(R\) and \(F\) in terms of \(T\). [4]
2. The coefficient of friction between the rod and the ring is 0.7. Find the value of \(T\) for which the ring is about to slip. [3]

\includegraphics{figure_5} 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 \(AP\) and \(BP\) of the string are taut. The system is in equilibrium with angle \(BAP = \alpha\) and angle \(ABP = \beta\) (see diagram). The weight of \(A\) is \(2\) N and the tensions in the parts \(AP\) and \(BP\) of the string are \(7\) N and \(T\) 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 \(A\). [7]
2. By considering the forces acting at \(P\), show that \(T \cos \beta = 1.96\). [2]
3. Given that there is no frictional force acting on \(B\), find the mass of \(B\). [3]