3.03e Resolve forces: two dimensions

\includegraphics{figure_5} A particle of mass 0.8 kg lies on a rough plane which is inclined at an angle of \(28°\) to the horizontal. The particle is kept in equilibrium by a force of magnitude \(T\) N. This force acts at an angle of \(35°\) above a line of greatest slope of the plane (see diagram). The coefficient of friction between the particle and the plane is 0.2. Find the least and greatest possible values of \(T\). [8]

\includegraphics{figure_6} A block \(B\), of mass 2 kg, lies on a rough inclined plane sloping at \(30°\) to the horizontal. A light rope, inclined at an angle of \(20°\) above a line of greatest slope, is attached to \(B\). The tension in the rope is \(T\) N. There is a friction force of \(F\) N acting on \(B\) (see diagram). The coefficient of friction between \(B\) and the plane is \(\mu\).

1. It is given that \(F = 5\) and that the acceleration of \(B\) up the plane is \(1.2\,\text{m}\,\text{s}^{-2}\).
   1. Find the value of \(T\). [3]
   2. Find the value of \(\mu\). [3]
2. It is given instead that \(\mu = 0.8\) and \(T = 15\). Determine whether \(B\) will move up the plane. [3]

\includegraphics{figure_4} A block of mass 8 kg is placed on a rough plane which is inclined at an angle of 18° to the horizontal. The block is pulled up the plane by a light string that makes an angle of 26° above a line of greatest slope. The tension in the string is \(T\) N (see diagram). The coefficient of friction between the block and plane is 0.65.

1. The acceleration of the block is 0.2 m s\(^{-2}\). Find \(T\). [7]
2. The block is initially at rest. Find the distance travelled by the block during the fourth second of motion. [2]

\includegraphics{figure_3} Forces of magnitudes 7 N, 10 N and 15 N act on a particle in the directions shown in the diagram.

1. Find the component of the resultant of the three forces
   1. in the \(x\)-direction,
   2. in the \(y\)-direction.
   [3]
2. Hence find the direction of the resultant. [2]

\includegraphics{figure_4} A block of mass 8 kg is at rest on a plane inclined at 20° to the horizontal. The block is connected to a vertical wall at the top of the plane by a string. The string is taut and parallel to a line of greatest slope of the plane (see diagram).

1. Given that the tension in the string is 13 N, find the frictional and normal components of the force exerted on the block by the plane. [4]

The string is cut; the block remains at rest, but is on the point of slipping down the plane.

1. Find the coefficient of friction between the block and the plane. [2]

\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_4} Coplanar forces of magnitudes 250 N, 160 N and 370 N act at a point \(O\) in the directions shown in the diagram, where the angle \(\alpha\) is such that \(\sin \alpha = 0.28\) and \(\cos \alpha = 0.96\). Calculate the magnitude of the resultant of the three forces. Calculate also the angle that the resultant makes with the \(x\)-direction. [7]

\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_4} Coplanar forces of magnitudes \(250 \text{ N}\), \(160 \text{ N}\) and \(370 \text{ N}\) act at a point \(O\) in the directions shown in the diagram, where the angle \(\alpha\) is such that \(\sin \alpha = 0.28\) and \(\cos \alpha = 0.96\). Calculate the magnitude of the resultant of the three forces. Calculate also the angle that the resultant makes with the \(x\)-direction. [7]

\includegraphics{figure_3} A particle is moving under the action of three forces as shown in the diagram. The particle is in equilibrium. Find the magnitudes of forces \(P\) and \(Q\). [6]

\includegraphics{figure_1} Four coplanar forces of magnitudes 4 N, 8 N, 12 N and 16 N act at a point. The directions in which the forces act are shown in Fig. 1.

1. Find the magnitude and direction of the resultant of the four forces. [5]

\includegraphics{figure_2} The forces of magnitudes 4 N and 16 N exchange their directions and the forces of magnitudes 8 N and 12 N also exchange their directions (see Fig. 2).

1. State the magnitude and direction of the resultant of the four forces in Fig. 2. [2]

\includegraphics{figure_5} A particle of mass \(0.12\) kg is placed on a plane which is inclined at an angle of \(40°\) to the horizontal. The particle is kept in equilibrium by a force of magnitude \(P\) N acting up the plane at an angle of \(30°\) above a line of greatest slope, as shown in the diagram. The coefficient of friction between the particle and the plane is \(0.32\). Find the set of possible values of \(P\). [8]

\includegraphics{figure_2} The diagram shows three coplanar forces acting at the point \(O\). The magnitudes of the forces are \(6 \text{ N}\), \(8 \text{ N}\) and \(10 \text{ N}\). The angle between the \(6 \text{ N}\) force and the \(8 \text{ N}\) force is \(90°\). The forces are in equilibrium. Find the other angles between the forces. [4]

\includegraphics{figure_3} A particle \(P\) of mass \(8 \text{ kg}\) is on a smooth plane inclined at an angle of \(30°\) to the horizontal. A force of magnitude \(100 \text{ N}\), making an angle of \(\theta°\) with a line of greatest slope and lying in the vertical plane containing the line of greatest slope, acts on \(P\) (see diagram).

1. Given that \(P\) is in equilibrium, show that \(\theta = 66.4\), correct to \(1\) decimal place, and find the normal reaction between the plane and \(P\). [4]
2. Given instead that \(\theta = 30\), find the acceleration of \(P\). [2]

\includegraphics{figure_3} Coplanar forces of magnitudes 8 N, 12 N and 18 N act at a point in the directions shown in the diagram. Find the magnitude and direction of the single additional force acting at the same point which will produce equilibrium. [6]

\includegraphics{figure_1} Coplanar forces of magnitudes 40 N, 32 N, \(P\) N and 17 N act at a point in the directions shown in the diagram. The system is in equilibrium. Find the values of \(P\) and \(\theta\). [6]

\includegraphics{figure_2} A particle \(P\) of mass \(1.6\) kg is suspended in equilibrium by two light inextensible strings attached to points \(A\) and \(B\). The strings make angles of \(20°\) and \(40°\) respectively with the horizontal (see diagram). Find the tensions in the two strings. [6]

\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_3} Four coplanar forces of magnitudes \(F\) N, \(5\) N, \(25\) N and \(15\) N are acting at a point \(P\) in the directions shown in the diagram. Given that the forces are in equilibrium, find the values of \(F\) and \(α\). [6]

\includegraphics{figure_3} A particle is in equilibrium on a smooth horizontal table when acted on by the three horizontal forces shown in the diagram.

1. Find the values of \(F\) and \(\theta\). [4]
2. The force of magnitude 7 N is now removed. State the magnitude and direction of the resultant of the remaining two forces. [2]

\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_7} A rough inclined plane of length 65 cm is fixed with one end at a height of 16 cm above the other end. Particles \(P\) and \(Q\), of masses \(0.13\) kg and \(0.11\) kg respectively, are attached to the ends of a light inextensible string which passes over a small smooth pulley at the top of the plane. Particle \(P\) is held at rest on the plane and particle \(Q\) hangs vertically below the pulley (see diagram). The system is released from rest and \(P\) starts to move up the plane.

1. Draw a diagram showing the forces acting on \(P\) during its motion up the plane. [1]
2. Show that \(T - F > 0.32\), where \(T\) N is the tension in the string and \(F\) N is the magnitude of the frictional force on \(P\). [4]

The coefficient of friction between \(P\) and the plane is 0.6.

1. Find the acceleration of \(P\). [6]

\includegraphics{figure_2} A block of mass 15 kg hangs in equilibrium below a horizontal ceiling attached to two strings as shown in the diagram. One of the strings is inclined at \(45°\) to the horizontal and the tension in this string is 120 N. The other string is inclined at \(θ°\) to the horizontal and the tension in this string is \(T\) N. Find the values of \(T\) and \(θ\). [6]

A smooth ring \(R\) of mass \(m\) kg is threaded on a light inextensible string \(ARB\). The ends of the string are attached to fixed points \(A\) and \(B\) with \(A\) vertically above \(B\). The string is taut and angle \(ARB = 90°\). The angle between the part \(AR\) of the string and the vertical is \(45°\). The ring is held in equilibrium in this position by a force of magnitude \(2.5\) N, acting on the ring in the direction \(BR\) (see diagram). Calculate the tension in the string and the mass of the ring. [4] \includegraphics{figure_1}

\includegraphics{figure_3} Particles \(P\) and \(Q\) have masses \(0.8\) kg and \(0.4\) kg respectively. \(P\) is attached to a fixed point \(A\) by a light inextensible string which is inclined at an angle \(\alpha°\) to the vertical. \(Q\) is attached to a fixed point \(B\), which is vertically below \(A\), by a light inextensible string of length \(0.3\) m. The string \(BQ\) is horizontal. \(P\) and \(Q\) are joined to each other by a light inextensible string which is vertical. The particles rotate in horizontal circles of radius \(0.3\) m about the axis through \(A\) and \(B\) with constant angular speed \(5\) rad s\(^{-1}\) (see diagram).

1. By considering the motion of \(Q\), find the tensions in the strings \(PQ\) and \(BQ\). [3]
2. Find the tension in the string \(AP\) and the value of \(\alpha\). [5]