3.03e Resolve forces: two dimensions

1. \begin{figure}[h] \end{figure} A particle \(P\) is attached to one end of a light inextensible string. The other end of the string is attached to a fixed point \(O\). A horizontal force of magnitude 12 N is applied to \(P\). The particle \(P\) is in equilibrium with the string taut and \(O P\) making an angle of \(20 ^ { \circ }\) with the downward vertical, as shown in Figure 1. Find

1. the tension in the string,
2. the weight of \(P\).

7. \begin{figure}[h] \end{figure} A package of mass 4 kg lies on a rough plane inclined at \(30 ^ { \circ }\) to the horizontal. The package is held in equilibrium by a force of magnitude 45 N acting at an angle of \(50 ^ { \circ }\) to the plane, as shown in Figure 3. The force is acting in a vertical plane through a line of greatest slope of the plane. The package is in equilibrium on the point of moving up the plane. The package is modelled as a particle. Find

1. the magnitude of the normal reaction of the plane on the package,
2. the coefficient of friction between the plane and the package.

3. \begin{figure}[h] \end{figure} A box of mass 5 kg lies on a rough plane inclined at \(30 ^ { \circ }\) to the horizontal. The box is held in equilibrium by a horizontal force of magnitude 20 N , as shown in Figure 2. The force acts in a vertical plane containing a line of greatest slope of the inclined plane.
The box is in equilibrium and on the point of moving down the plane. The box is modelled as a particle. Find

1. the magnitude of the normal reaction of the plane on the box,
2. the coefficient of friction between the box and the plane.

1. \begin{figure}[h] \end{figure} A particle \(P\) of weight \(W\) newtons is attached to one end of a light inextensible string. The other end of the string is attached to a fixed point \(O\). A horizontal force of magnitude 5 N is applied to \(P\). The particle \(P\) is in equilibrium with the string taut and with \(O P\) making an angle of \(25 ^ { \circ }\) to the downward vertical, as shown in Figure 1. Find

1. the tension in the string,
2. the value of \(W\).

7. \begin{figure}[h] \end{figure} A particle \(P\) of mass 2.7 kg lies on a rough plane inclined at \(40 ^ { \circ }\) to the horizontal. The particle is held in equilibrium by a force of magnitude 15 N acting at an angle of \(50 ^ { \circ }\) to the plane, as shown in Figure 4. The force acts in a vertical plane containing a line of greatest slope of the plane. The particle is in equilibrium and is on the point of sliding down the plane. Find

1. the magnitude of the normal reaction of the plane on \(P\),
2. the coefficient of friction between \(P\) and the plane. The force of magnitude 15 N is removed.
3. Determine whether \(P\) moves, justifying your answer.

1. \begin{figure}[h] \end{figure} A particle of weight \(W\) newtons is attached at \(C\) to two light inextensible strings \(A C\) and \(B C\). The other ends of the strings are attached to fixed points \(A\) and \(B\) on a horizontal ceiling. The particle hangs in equilibrium with \(A C\) and \(B C\) inclined to the horizontal at \(30 ^ { \circ }\) and \(50 ^ { \circ }\) respectively, as shown in Figure 1. Given that the tension in \(B C\) is 6 N , find

1. the tension in \(A C\),
2. the value of \(W\).

3. \begin{figure}[h] \end{figure} A particle of mass 2 kg is suspended from a horizontal ceiling by two light inextensible strings, \(P R\) and \(Q R\). The particle hangs at \(R\) in equilibrium, with the strings in a vertical plane. The string \(P R\) is inclined at \(55 ^ { \circ }\) to the horizontal and the string \(Q R\) is inclined at \(35 ^ { \circ }\) to the horizontal, as shown in Figure 1. \section*{Find}

1. the tension in the string \(P R\),
2. the tension in the string \(Q R\).

5. \begin{figure}[h] \end{figure} A particle \(P\) of mass 2 kg is held at rest in equilibrium on a rough plane by a constant force of magnitude 40 N . The direction of the force is inclined to the plane at an angle of \(30 ^ { \circ }\). The plane is inclined to the horizontal at an angle of \(20 ^ { \circ }\), as shown in Figure 2. The line of action of the force lies in the vertical plane containing \(P\) and a line of greatest slope of the plane. The coefficient of friction between \(P\) and the plane is \(\mu\). Given that \(P\) is on the point of sliding up the plane, find the value of \(\mu\).

8. \begin{figure}[h] \end{figure} Two particles, \(A\) and \(B\), have masses 2 kg and 4 kg respectively. The particles are connected by a light inextensible string. The string passes over a small smooth pulley which is fixed at the top of a rough plane. The plane is inclined to the horizontal ground at an angle \(\alpha\) where \(\tan \alpha = \frac { 3 } { 4 }\). The particle \(A\) is held at rest on the plane at a distance \(d\) metres from the pulley. The particle \(B\) hangs freely at rest, vertically below the pulley, at a distance \(h\) metres above the ground, as shown in Figure 3. The part of the string between \(A\) and the pulley is parallel to a line of greatest slope of the plane. The coefficient of friction between \(A\) and the plane is \(\frac { 1 } { 4 }\) The system is released from rest with the string taut and \(B\) descends.

1. Find the tension in the string as \(B\) descends. On hitting the ground, \(B\) immediately comes to rest. Given that \(A\) comes to rest before reaching the pulley,
2. find, in terms of \(h\), the range of possible values of \(d\).
3. State one physical factor, other than air resistance, that could be taken into account to make the model described above more realistic.

5. \begin{figure}[h] \end{figure} A particle of mass \(m\) rests in equilibrium on a fixed rough plane under the action of a force of magnitude \(X\). The force acts up a line of greatest slope of the plane, as shown in Figure 3. The plane 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 \(\mu\).

• When \(X = 2 P\), the particle is on the point of sliding up the plane.
• When \(X = P\), the particle is on the point of sliding down the plane.

Find the value of \(\mu\).

7. \begin{figure}[h] \end{figure} A particle \(P\) of mass \(4 m\) lies on the surface of a fixed rough inclined plane.
The plane is inclined to the horizontal at an angle \(\alpha\) where \(\tan \alpha = \frac { 3 } { 4 }\) The particle \(P\) is attached to one end of a light inextensible string.
The string passes over a small smooth pulley that is fixed at the top of the plane. The other end of the string is attached to a particle \(Q\) of mass \(m\) which lies on a smooth horizontal plane. The string lies along the horizontal plane and in the vertical plane that contains the pulley and a line of greatest slope of the inclined plane. The system is released from rest with the string taut, as shown in Figure 4, and \(P\) moves down the plane. The coefficient of friction between \(P\) and the plane is \(\frac { 1 } { 4 }\) For the motion before \(Q\) reaches the pulley

1. write down an equation of motion for \(Q\),
2. find, in terms of \(m\) and \(g\), the tension in the string,
3. find the magnitude of the force exerted on the pulley by the string.
4. State where in your working you have used the information that the string is light.

6. \begin{figure}[h] \end{figure} A boat is pulled along a river at a constant speed by two ropes.
The banks of the river are parallel and the boat travels horizontally in a straight line, parallel to the riverbanks.

• The tension in the first rope is 500 N acting at an angle of \(40 ^ { \circ }\) to the direction of motion, as shown in Figure 3.
• The tension in the second rope is \(P\) newtons, acting at an angle of \(\alpha ^ { \circ }\) to the direction of motion, also shown in Figure 3.
• The resistance to motion of the boat as it moves through the water is a constant force of magnitude 900 N

The boat is modelled as a particle. The ropes are modelled as being light and lying in a horizontal plane. Use the model to find

1. the value of \(\alpha\)
2. the value of \(P\)

8. \begin{figure}[h] \end{figure} A parcel of mass 2 kg is pulled up a rough inclined plane by the action of a constant force. The force has magnitude 18 N and acts at an angle of \(40 ^ { \circ }\) to the plane.
The line of action of the force lies in a vertical plane containing a line of greatest slope of the inclined plane. The plane is inclined at an angle of \(30 ^ { \circ }\) to the horizontal, as shown in Figure 5.
The coefficient of friction between the plane and the parcel is 0.3
The parcel is modelled as a particle \(P\)

1. Find the acceleration of \(P\) The points \(A\) and \(B\) lie on a line of greatest slope of the plane, where \(A B = 5 \mathrm {~m}\) and \(B\) is above \(A\). Particle \(P\) passes through \(A\) with speed \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the direction \(A B\).
2. Find the speed of \(P\) as it passes through \(B\). The force of 18 N is removed at the instant \(P\) passes through \(B\). As a result, \(P\) comes to rest at the point \(C\).
3. Determine whether \(P\) will remain at rest at \(C\). You must show all stages of your working clearly.

3. A particle \(P\) of mass 1.5 kg is placed at a point \(A\) on a rough plane which is inclined at \(30 ^ { \circ }\) to the horizontal. The coefficient of friction between \(P\) and the plane is 0.6

1. Show that \(P\) rests in equilibrium at \(A\). A horizontal force of magnitude \(X\) newtons is now applied to \(P\), as shown in Figure 1. The force acts in a vertical plane containing a line of greatest slope of the inclined plane. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{edcc4603-f006-4c4f-a4e5-063cab41da98-04_236_584_667_680} \captionsetup{labelformat=empty} \caption{Figure 1}
   \end{figure} The particle is on the point of moving up the plane.
2. Find
   1. the magnitude of the normal reaction of the plane on \(P\),
   2. the value of \(X\).

5. \begin{figure}[h] \end{figure} A particle \(P\) of mass 0.5 kg is at rest on a rough plane which is inclined to the horizontal at \(30 ^ { \circ }\). The particle is held in equilibrium by a force of magnitude 8 N , acting at an angle of \(40 ^ { \circ }\) to the plane, as shown in Figure 2. The line of action of the force lies in the vertical plane containing \(P\) and a line of greatest slope of the plane. The coefficient of friction between \(P\) and the plane is \(\mu\). Given that \(P\) is on the point of sliding up the plane, find the value of \(\mu\).

8. \begin{figure}[h] \end{figure} A rough plane is inclined at \(30 ^ { \circ }\) to the horizontal. A particle \(P\) of mass 0.5 kg is held at rest on the plane by a horizontal force of magnitude 5 N , as shown in Figure 4. The force acts in a vertical plane containing a line of greatest slope of the inclined plane. The particle is on the point of moving up the plane.

1. Find the magnitude of the normal reaction of the plane on \(P\).
2. Find the coefficient of friction between \(P\) and the plane. The force of magnitude 5 N is now removed and \(P\) accelerates from rest down the plane.
3. Find the speed of \(P\) after it has travelled 3 m down the plane.

7. \begin{figure}[h] \end{figure} A particle \(P\) of mass 4 kg is attached to one end of a light inextensible string. A particle \(Q\) of mass \(m \mathrm {~kg}\) is attached to the other end of the string. The string passes over a small smooth pulley which is fixed at a point on the intersection of two fixed inclined planes. The string lies in a vertical plane that contains a line of greatest slope of each of the two inclined planes. The first plane is inclined to the horizontal at an angle \(\alpha\), where \(\tan \alpha = \frac { 3 } { 4 }\) and the second plane is inclined to the horizontal at an angle \(\beta\), where \(\tan \beta = \frac { 4 } { 3 }\). Particle \(P\) is on the first plane and particle \(Q\) is on the second plane with the string taut, as shown in Figure 3. The first plane is rough and the coefficient of friction between \(P\) and the plane is \(\frac { 1 } { 4 }\). The second plane is smooth. The system is in limiting equilibrium. Given that \(P\) is on the point of slipping down the first plane,

1. find the value of \(m\),
2. find the magnitude of the force exerted on the pulley by the string,
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2. \begin{figure}[h] \end{figure} A particle \(P\) of mass 2 kg is held in equilibrium under gravity by two light inextensible strings. One string is horizontal and the other is inclined at an angle \(\alpha\) to the horizontal, as shown in Fig. 2. The tension in the horizontal string is 15 N . The tension in the other string is \(T\) newtons.

1. Find the size of the angle \(\alpha\).
2. Find the value of \(T\).

7. A sledge of mass 78 kg is pulled up a slope by means of a rope. The slope is modelled as a rough plane inclined at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac { 5 } { 12 }\). The rope is modelled as light and inextensible and is in a line of greatest slope of the plane. The coefficient of friction between the sledge and the slope is 0.25 . Given that the sledge is accelerating up the slope with acceleration \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\),

1. find the tension in the rope. The rope suddenly breaks. Subsequently the sledge comes to instantaneous rest and then starts sliding down the slope.
2. Find the acceleration of the sledge down the slope after it has come to instantaneous rest.
   (6 marks)
   END

4. \begin{figure}[h] \end{figure} A particle \(P\) of mass 6 kg lies on the surface of a smooth plane. The plane is inclined at an angle of \(30 ^ { \circ }\) to the horizontal. The particle is held in equilibrium by a force of magnitude 49 N , acting at an angle \(\theta\) to the plane, as shown in Figure 1. The force acts in a vertical plane through a line of greatest slope of the plane.

1. Show that \(\cos \theta = \frac { 3 } { 5 }\).
2. Find the normal reaction between \(P\) and the plane. The direction of the force of magnitude 49 N is now changed. It is now applied horizontally to \(P\) so that \(P\) moves up the plane. The force again acts in a vertical plane through a line of greatest slope of the plane.
3. Find the initial acceleration of \(P\). \(\_\_\_\_\)}

5. \section*{Figure 2} A small package of mass 1.1 kg is held in equilibrium on a rough plane by a horizontal force. The plane is inclined at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac { 3 } { 4 }\). The force acts in a vertical plane containing a line of greatest slope of the plane and has magnitude \(P\) newtons, as shown in Figure 2. The coefficient of friction between the package and the plane is 0.5 and the package is modelled as a particle. The package is in equilibrium and on the point of slipping down the plane.

1. Draw, on Figure 2, all the forces acting on the package, showing their directions clearly.
2. 1. Find the magnitude of the normal reaction between the package and the plane.
   2. Find the value of \(P\).

6. Two forces, \(( 4 \mathbf { i } - 5 \mathbf { j } ) \mathrm { N }\) and \(( p \mathbf { i } + q \mathbf { j } ) \mathrm { N }\), act on a particle \(P\) of mass \(m \mathrm {~kg}\). The resultant of the two forces is \(\mathbf { R }\). Given that \(\mathbf { R }\) acts in a direction which is parallel to the vector ( \(\mathbf { i } - 2 \mathbf { j }\) ),

1. find the angle between \(\mathbf { R }\) and the vector \(\mathbf { j }\),
2. show that \(2 p + q + 3 = 0\). Given also that \(q = 1\) and that \(P\) moves with an acceleration of magnitude \(8 \sqrt { } 5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), (c) find the value of \(m\).

3 \includegraphics[max width=\textwidth, alt={}, center]{470e70de-66ba-4dcc-a205-0c92f29471b1-2_570_679_1512_731} Three horizontal forces act at the point \(O\). One force has magnitude 7 N and acts along the positive \(x\)-axis. The second force has magnitude 9 N and acts along the positive \(y\)-axis. The third force has magnitude 5 N and acts at an angle of \(30 ^ { \circ }\) below the negative \(x\)-axis (see diagram).

1. Find the magnitudes of the components of the 5 N force along the two axes.
2. Calculate the magnitude of the resultant of the three forces. Calculate also the angle the resultant makes with the positive \(x\)-axis.

4 \includegraphics[max width=\textwidth, alt={}, center]{470e70de-66ba-4dcc-a205-0c92f29471b1-3_200_897_269_625} A block of mass 3 kg is placed on a horizontal surface. A force of magnitude 20 N acts downwards on the block at an angle of \(30 ^ { \circ }\) to the horizontal (see diagram).

1. Given that the surface is smooth, calculate the acceleration of the block.
2. Given instead that the block is in limiting equilibrium, calculate the coefficient of friction between the block and the surface.

1
A light inextensible string has its ends attached to two fixed points \(A\) and \(B\). The point \(A\) is vertically above \(B\). A smooth ring \(R\) of mass \(m \mathrm {~kg}\) is threaded on the string and is pulled by a force of magnitude 1.6 N acting upwards at \(45 ^ { \circ }\) to the horizontal. The section \(A R\) of the string makes an angle of \(30 ^ { \circ }\) with the downward vertical and the section \(B R\) is horizontal (see diagram). The ring is in equilibrium with the string taut.

1. Give a reason why the tension in the part \(A R\) of the string is the same as that in the part \(B R\).
2. Show that the tension in the string is 0.754 N , correct to 3 significant figures.
3. Find the value of \(m\).