3.03e Resolve forces: two dimensions

4 \includegraphics[max width=\textwidth, alt={}, center]{9fbb63e3-4017-461e-9110-500be2c20778-2_583_862_1343_644} Three coplanar forces of magnitudes \(68 \mathrm {~N} , 75 \mathrm {~N}\) and 100 N act at an origin \(O\), as shown in the diagram. The components of the three forces in the positive \(x\)-direction are \(- 60 \mathrm {~N} , 0 \mathrm {~N}\) and 96 N , respectively. Find

1. the components of the three forces in the positive \(y\)-direction,
2. the magnitude and direction of the resultant of the three forces. \(5 A , B\) and \(C\) are three points on a line of greatest slope of a plane which is inclined at \(\theta ^ { \circ }\) to the horizontal, with \(A\) higher than \(B\) and \(B\) higher than \(C\). Between \(A\) and \(B\) the plane is smooth, and between \(B\) and \(C\) the plane is rough. A particle \(P\) is released from rest on the plane at \(A\) and slides down the line \(A B C\). At time 0.8 s after leaving \(A\), the particle passes through \(B\) with speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1 \includegraphics[max width=\textwidth, alt={}, center]{3e58aa5a-3789-4aaf-8656-b5b98cd7f693-2_291_591_255_776} A particle \(P\) of mass 0.3 kg is attached to one end of a light inextensible string. The other end of the string is attached to a fixed point \(X\). A horizontal force of magnitude \(F \mathrm {~N}\) is applied to the particle, which is in equilibrium when the string is at an angle \(\alpha\) to the vertical, where \(\tan \alpha = \frac { 8 } { 15 }\) (see diagram). Find the tension in the string and the value of \(F\).

2 \includegraphics[max width=\textwidth, alt={}, center]{3e58aa5a-3789-4aaf-8656-b5b98cd7f693-2_385_389_918_879} A block \(B\) lies on a rough horizontal plane. Horizontal forces of magnitudes 30 N and 40 N , making angles of \(\alpha\) and \(\beta\) respectively with the \(x\)-direction, act on \(B\) as shown in the diagram, and \(B\) is moving in the \(x\)-direction with constant speed. It is given that \(\cos \alpha = 0.6\) and \(\cos \beta = 0.8\).

1. Find the total work done by the forces shown in the diagram when \(B\) has moved a distance of 20 m .
2. Given that the coefficient of friction between the block and the plane is \(\frac { 5 } { 8 }\), find the weight of the block.

1 \includegraphics[max width=\textwidth, alt={}, center]{79b90ef5-ef3a-4c59-b662-d0fbfba813ca-2_346_583_255_781} A small block of weight 5.1 N rests on a smooth plane inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 8 } { 17 }\). The block is held in equilibrium by means of a light inextensible string. The string makes an angle \(\beta\) above the line of greatest slope on which the block rests, where \(\sin \beta = \frac { 7 } { 25 }\) (see diagram). Find the tension in the string.

4 \includegraphics[max width=\textwidth, alt={}, center]{79b90ef5-ef3a-4c59-b662-d0fbfba813ca-2_365_493_1749_826} A rough plane is inclined at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = 2.4\). A small block of mass 0.6 kg is held at rest on the plane by a horizontal force of magnitude \(P \mathrm {~N}\). This force acts in a vertical plane through a line of greatest slope (see diagram). The coefficient of friction between the block and the plane is 0.4 . The block is on the point of slipping down the plane. By resolving forces parallel to and perpendicular to the inclined plane, or otherwise, find the value of \(P\).
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3 \includegraphics[max width=\textwidth, alt={}, center]{fd534430-2619-4078-ad0a-2355e656e121-2_307_857_1695_644} A particle \(P\) of mass 1.05 kg is attached to one end of each of two light inextensible strings, of lengths 2.6 m and 1.25 m . The other ends of the strings are attached to fixed points \(A\) and \(B\), which are at the same horizontal level. \(P\) hangs in equilibrium at a point 1 m below the level of \(A\) and \(B\) (see diagram). Find the tensions in the strings.

2 \includegraphics[max width=\textwidth, alt={}, center]{ffefbc81-402f-4048-8741-23c8bae30d5a-2_385_621_488_762} Small blocks \(A\) and \(B\) are held at rest on a smooth plane inclined at \(30 ^ { \circ }\) to the horizontal. Each is held in equilibrium by a force of magnitude 18 N . The force on \(A\) acts upwards parallel to a line of greatest slope of the plane, and the force on \(B\) acts horizontally in the vertical plane containing a line of greatest slope (see diagram). Find the weight of \(A\) and the weight of \(B\).

2 \includegraphics[max width=\textwidth, alt={}, center]{c7133fc4-9a14-43fd-b5ed-788da72291cd-2_666_953_662_596} Three coplanar forces act at a point. The magnitudes of the forces are \(20 \mathrm {~N} , 25 \mathrm {~N}\) and 30 N , and the directions in which the forces act are as shown in the diagram, where \(\sin \alpha = 0.28\) and \(\cos \alpha = 0.96\), and \(\sin \beta = 0.6\) and \(\cos \beta = 0.8\).

1. Show that the resultant of the three forces has a zero component in the \(x\)-direction.
2. Find the magnitude and direction of the resultant of the three forces.
3. The force of magnitude 20 N is replaced by another force. The effect is that the resultant force is unchanged in magnitude but reversed in direction. State the magnitude and direction of the replacement force.

5 \includegraphics[max width=\textwidth, alt={}, center]{2a91fb7a-0eaf-4c50-8a2c-4755c0b44c17-3_355_1048_255_552} A small bead \(Q\) can move freely along a smooth horizontal straight wire \(A B\) of length 3 m . Three horizontal forces of magnitudes \(F \mathrm {~N} , 10 \mathrm {~N}\) and 20 N act on the bead in the directions shown in the diagram. The magnitude of the resultant of the three forces is \(R \mathrm {~N}\) in the direction shown in the diagram.

1. Find the values of \(F\) and \(R\).
2. Initially the bead is at rest at \(A\). It reaches \(B\) with a speed of \(11.7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the mass of the bead.

1 \includegraphics[max width=\textwidth, alt={}, center]{48f66bd5-33c1-4ce9-85f9-69faf10e871c-2_558_529_258_808} Four horizontal forces act at a point \(O\) and are in equilibrium. The magnitudes of the forces are \(F \mathrm {~N}\), \(G \mathrm {~N} , 15 \mathrm {~N}\) and \(F \mathrm {~N}\), and the forces act in directions as shown in the diagram.

1. Show that \(F = 41.0\), correct to 3 significant figures.
2. Find the value of \(G\).

2 \includegraphics[max width=\textwidth, alt={}, center]{94c11160-a718-4de5-867a-27c755051fa6-2_342_629_616_756} The diagram shows a small object \(P\) of mass 20 kg held in equilibrium by light ropes attached to fixed points \(A\) and \(B\). The rope \(P A\) is inclined at an angle of \(50 ^ { \circ }\) above the horizontal, the rope \(P B\) is inclined at an angle of \(10 ^ { \circ }\) below the horizontal, and both ropes are in the same vertical plane. Find the tension in the rope \(P A\) and the tension in the rope \(P B\).

1 A block of mass 3 kg is initially at rest on a smooth horizontal floor. A force of 12 N , acting at an angle of \(25 ^ { \circ }\) above the horizontal, is applied to the block. Find the distance travelled by the block in the first 5 seconds of its motion.

5 \includegraphics[max width=\textwidth, alt={}, center]{98a5537b-d503-4a42-bbfe-0bd221084ee0-06_449_654_260_742} Coplanar forces, of magnitudes \(15 \mathrm {~N} , 25 \mathrm {~N}\) and 30 N , act at a point \(B\) on the line \(A B C\) in the directions shown in the diagram.

1. Find the magnitude and direction of the resultant force.
2. The force of magnitude 15 N is now replaced by a force of magnitude \(F \mathrm {~N}\) acting in the same direction. The new resultant force has zero component in the direction \(B C\). Find the value of \(F\), and find also the magnitude and direction of the new resultant force.

5
A toy aircraft of mass 0.5 kg is attached to one end of a light inextensible string of length 9 m . The other end of the string is attached to a fixed point \(O\). The aircraft moves with constant speed in a horizontal circle. The string is taut, and makes an angle of \(60 ^ { \circ }\) with the upward vertical at \(O\) (see diagram). In a simplified model of the motion, the aircraft is treated as a particle and the force of the air on the aircraft is taken to act vertically upwards with magnitude 8 N . Find

1. the tension in the string,
2. the speed of the aircraft.

3 \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure} A hollow container consists of a smooth circular cylinder of radius 0.5 m , and a smooth hollow cone of semi-vertical angle \(65 ^ { \circ }\) and radius 0.5 m . The container is fixed with its axis vertical and with the cone below the cylinder. A steel ball of weight 1 N moves with constant speed \(2.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a horizontal circle inside the container. The ball is in contact with both the cylinder and the cone (see Fig. 1). Fig. 2 shows the forces acting on the ball, i.e. its weight and the forces of magnitudes \(R \mathrm {~N}\) and \(S \mathrm {~N}\) exerted by the container at the points of contact. Given that the radius of the ball is negligible compared with the radius of the cylinder, find \(R\) and \(S\).

3 \includegraphics[max width=\textwidth, alt={}, center]{36259e2a-aa9b-4655-b0c2-891f96c3f5a4-3_637_572_264_788} One end of a light inextensible string is attached to a point \(C\). The other end is attached to a point \(D\), which is 1.1 m vertically below \(C\). A small smooth ring \(R\), of mass 0.2 kg , is threaded on the string and moves with constant speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a horizontal circle, with centre at \(O\) and radius 1.2 m , where \(O\) is 0.5 m vertically below \(D\) (see diagram).

1. Show that the tension in the string is 1.69 N , correct to 3 significant figures.
2. Find the value of \(v\).

4
A particle of mass 0.12 kg is moving on the smooth inside surface of a fixed hollow sphere of radius 0.5 m . The particle moves in a horizontal circle whose centre is 0.3 m below the centre of the sphere (see diagram).

1. Show that the force exerted by the sphere on the particle has magnitude 2 N .
2. Find the speed of the particle.
3. Find the time taken for the particle to complete one revolution.

3 \includegraphics[max width=\textwidth, alt={}, center]{ae809dfc-c5af-4c0a-9c88-009949d3e9f9-3_456_511_260_817} A particle of mass 0.24 kg is attached to one end of a light inextensible string of length 2 m . The other end of the string is attached to a fixed point. The particle moves with constant speed in a horizontal circle. The string makes an angle \(\theta\) with the vertical (see diagram), and the tension in the string is \(T \mathrm {~N}\). The acceleration of the particle has magnitude \(7.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

1. Show that \(\tan \theta = 0.75\) and find the value of \(T\).
2. Find the speed of the particle.

5 \includegraphics[max width=\textwidth, alt={}, center]{9daebcbe-826e-4eda-afa7-c935c6ea2bfc-06_671_504_255_824} \(A\) and \(B\) are two fixed points on a vertical axis with \(A\) above \(B\). A particle \(P\) of mass 0.4 kg is attached to \(A\) by a light inextensible string of length 0.5 m . The particle \(P\) is attached to \(B\) by another light inextensible string. \(P\) moves with constant speed in a horizontal circle with centre \(O\) between \(A\) and \(B\). Angle \(B A P = 30 ^ { \circ }\) and angle \(A B P = 70 ^ { \circ }\) (see diagram).

1. Given that the tensions in the two strings are equal, find the speed of \(P\).
2. Given instead that the angular speed of \(P\) is \(12 \mathrm { rad } \mathrm { s } ^ { - 1 }\), find the tensions in the strings.

2 One end of a light inextensible string of length 0.5 m is attached to a fixed point \(A\). A particle \(P\) of mass 0.2 kg is attached to the other end of the string. \(P\) moves with constant speed in a horizontal circle with centre \(O\) which is 0.4 m vertically below \(A\).

1. Show that the tension in the string is 2.5 N .
2. Find the speed of \(P\).

4 \includegraphics[max width=\textwidth, alt={}, center]{add3948c-3b45-4e67-ac84-e2ca935afd64-05_392_621_255_762} A particle \(P\) of mass 0.4 kg moves with constant speed in a horizontal circle on the smooth inner surface of a fixed hollow hemisphere with centre \(O\) and radius 0.5 m (see diagram).

1. Given that the speed of the particle is \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and its angular speed is \(10 \mathrm { rad } \mathrm { s } ^ { - 1 }\), calculate the angle between \(O P\) and the vertical.
2. Given instead that the magnitude of the force exerted on \(P\) by the hemisphere is 6 N , calculate
   1. the angle between \(O P\) and the vertical,
   2. the angular speed of \(P\).

6. \begin{figure}[h] \end{figure} A particle of weight 120 N is placed on a fixed rough plane which is inclined at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac { 3 } { 4 }\).
The coefficient of friction between the particle and the plane is \(\frac { 1 } { 2 }\).
The particle is held at rest in equilibrium by a horizontal force of magnitude 30 N , which acts in the vertical plane containing the line of greatest slope of the plane through the particle, as shown in Figure 2.

1. Show that the normal reaction between the particle and the plane has magnitude 114 N . \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{4878b6c2-0c62-4398-8a8f-913139bc8a14-10_433_774_1464_604} \captionsetup{labelformat=empty} \caption{Figure 3}
   \end{figure} The horizontal force is removed and replaced by a force of magnitude \(P\) newtons acting up the slope along the line of greatest slope of the plane through the particle, as shown in Figure 3. The particle remains in equilibrium.
2. Find the greatest possible value of \(P\).
3. Find the magnitude and direction of the frictional force acting on the particle when \(P = 30\).

8. \begin{figure}[h] \end{figure} A particle \(P\) of mass 4 kg is moving up a fixed rough plane at a constant speed of \(16 \mathrm {~ms} ^ { - 1 }\) under the action of a force of magnitude 36 N . The plane is inclined at \(30 ^ { \circ }\) to the horizontal. The force acts in the vertical plane containing the line of greatest slope of the plane through \(P\), and acts at \(30 ^ { \circ }\) to the inclined plane, as shown in Figure 2. The coefficient of friction between \(P\) and the plane is \(\mu\). Find

1. the magnitude of the normal reaction between \(P\) and the plane,
2. the value of \(\mu\). The force of magnitude 36 N is removed.
3. Find the distance that \(P\) travels between the instant when the force is removed and the instant when it comes to rest.

4. \begin{figure}[h] \end{figure} A small parcel of mass 3 kg is held in equilibrium on a rough plane by the action of a horizontal force of magnitude 30 N acting in a vertical plane through a line of greatest slope. The plane is inclined at an angle of \(30 ^ { \circ }\) to the horizontal, as shown in Fig. 3. The parcel is modelled as a particle. The parcel is on the point of moving up the slope.

1. Draw a diagram showing all the forces acting on the parcel.
2. Find the normal reaction on the parcel.
3. Find the coefficient of friction between the parcel and the plane.

4. \begin{figure}[h] \end{figure} A parcel of mass 5 kg lies on a rough plane inclined at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac { 3 } { 4 }\). The parcel is held in equilibrium by the action of a horizontal force of magnitude 20 N , as shown in Fig. 2. The force acts in a vertical plane through a line of greatest slope of the plane. The parcel is on the point of sliding down the plane. Find the coefficient of friction between the parcel and the plane.
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