Ring on horizontal rod equilibrium

3 A ring of mass 0.3 kg is threaded on a horizontal rough rod. The coefficient of friction between the ring and the rod is 0.8 . A force of magnitude 8 N acts on the ring. This force acts at an angle of \(10 ^ { \circ }\) above the horizontal in the vertical plane containing the rod. Find the time taken for the ring to move, from rest, 0.6 m along the rod.

1 \includegraphics[max width=\textwidth, alt={}, center]{e060fc3b-ae93-46b5-b476-dcecb14d6d06-2_200_588_267_781} A ring of mass 1.1 kg is threaded on a fixed rough horizontal rod. A light string is attached to the ring and the string is pulled with a force of magnitude 13 N at an angle \(\alpha\) below the horizontal, where \(\tan \alpha = \frac { 5 } { 12 }\) (see diagram). The ring is in equilibrium.

1. Find the frictional component of the contact force on the ring.
2. Find the normal component of the contact force on the ring.
3. Given that the equilibrium of the ring is limiting, find the coefficient of friction between the ring and the rod.

4 \includegraphics[max width=\textwidth, alt={}, center]{fa0e0e0d-b0a6-44e0-8b4f-4923e235c6c6-2_168_711_1612_717} A ring of mass 4 kg is attached to one end of a light string. The ring is threaded on a fixed horizontal rod and the string is pulled at an angle of \(25 ^ { \circ }\) below the horizontal (see diagram). With a tension in the string of \(T \mathrm {~N}\) the ring is in equilibrium.

1. Find, in terms of \(T\), the horizontal and vertical components of the force exerted on the ring by the rod. The coefficient of friction between the ring and the rod is 0.4 .
2. Given that the equilibrium is limiting, find the value of \(T\).

2 \includegraphics[max width=\textwidth, alt={}, center]{f23ea8e7-9b81-4192-8c20-8c46aabfecca-2_195_719_1114_712} A ring of mass 0.2 kg is threaded on a fixed rough horizontal rod and a light inextensible string is attached to the ring at an angle \(\alpha\) above the horizontal, where \(\cos \alpha = 0.96\). The ring is in limiting equilibrium with the tension in the string \(T \mathrm {~N}\) (see diagram). Given that the coefficient of friction between the ring and the rod is 0.25 , find the value of \(T\).
[0pt] [5]

5. \begin{figure}[h] \end{figure} A small ring of mass 0.25 kg is threaded on a fixed rough horizontal rod. The ring is pulled upwards by a light string which makes an angle \(40 ^ { \circ }\) with the horizontal, as shown in Figure 3. The string and the rod are in the same vertical plane. The tension in the string is 1.2 N and the coefficient of friction between the ring and the rod is \(\mu\). Given that the ring is in limiting equilibrium, find

1. the normal reaction between the ring and the rod,
2. the value of \(\mu\).

\includegraphics{figure_3} A small ring of mass 0.8 kg is threaded on a rough rod which is fixed horizontally. The ring is in equilibrium, acted on by a force of magnitude 7 N pulling upwards at 45° to the horizontal (see diagram).

1. Show that the normal component of the contact force acting on the ring has magnitude 3.05 N, correct to 3 significant figures. [2]
2. The ring is in limiting equilibrium. Find the coefficient of friction between the ring and the rod. [3]

\includegraphics{figure_3} A small ring of mass \(0.8 \text{ kg}\) is threaded on a rough rod which is fixed horizontally. The ring is in equilibrium, acted on by a force of magnitude \(7 \text{ N}\) pulling upwards at \(45°\) to the horizontal (see diagram).

1. Show that the normal component of the contact force acting on the ring has magnitude \(3.05 \text{ N}\), correct to 3 significant figures. [2]
2. The ring is in limiting equilibrium. Find the coefficient of friction between the ring and the rod. [3]

\includegraphics{figure_2} A ring of mass 0.3 kg is threaded on a fixed, rough horizontal curtain pole. A light inextensible string is attached to the ring. The string and the pole lie in the same vertical plane. The ring is pulled downwards by the string which makes an angle \(\alpha\) to the horizontal, where tan \(\alpha = \frac{3}{4}\) as shown in Fig. 2. The tension in the string is 2.5 N. Given that, in this position, the ring is in limiting equilibrium,

1. find the coefficient of friction between the ring and the pole. [8]

\includegraphics{figure_3} The direction of the string is now altered so that the ring is pulled upwards. The string lies in the same vertical plane as before and again makes an angle \(\alpha\) with the horizontal, as shown in Fig. 3. The tension in the string is again 2.5 N.

1. Find the normal reaction exerted by the pole on the ring. [2]
2. State whether the ring is in equilibrium in the position shown in Fig. 3, giving a brief justification for your answer. You need make no further detailed calculation of the forces acting. [2]