3.03e Resolve forces: two dimensions

5 A cylindrical tub of mass 250 kg is on a horizontal floor. Resistance to its motion other than that due to friction is negligible. The first attempt to move the tub is by pulling it with a force of 150 N in the \(\mathbf { i }\) direction, as shown in Fig. 8.1. \begin{figure}[h] \end{figure}

1. Calculate the acceleration of the tub if friction is ignored. In fact, there is friction and the tub does not move.
2. Write down the magnitude and direction of the frictional force opposing the pull. Two more forces are now added to the 150 N force in a second attempt to move the tub, as shown in Fig. 8.2. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{5a1895e1-abe3-4739-876a-f19458f0f6ed-4_497_927_1350_646} \captionsetup{labelformat=empty} \caption{Fig. 5.2}
   \end{figure} Angle \(\theta\) is acute and chosen so that the resultant of the three forces is in the \(\mathbf { i }\) direction.
3. Determine the value of \(\theta\) and the resultant of the three forces. With this resultant force, the tub moves with constant acceleration and travels 1 metre from rest in 2 seconds.
4. Show that the magnitude of the friction acting on the tub is 661 N , correct to 3 significant figures. When the speed of the tub is \(1.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), it comes to a part of the floor where the friction on the tub is 200 N greater. The pulling forces stay the same.
5. Find the velocity of the tub when it has moved a further 1.65 m .

6 An empty open box of mass 4 kg is on a plane that is inclined at \(25 ^ { \circ }\) to the horizontal.
In one model the plane is taken to be smooth. The box is held in equilibrium by a string with tension \(T \mathrm {~N}\) parallel to the plane, as shown in Fig. 6.1. \begin{figure}[h] \end{figure}

1. Calculate \(T\). A rock of mass \(m \mathrm {~kg}\) is now put in the box. The system is in equilibrium when the tension in the string, still parallel to the plane, is 50 N .
2. Find \(m\). In a refined model the plane is rough. The empty box, of mass 4 kg , is in equilibrium when a frictional force of 20 N acts down the plane and the string has a tension of \(P \mathrm {~N}\) inclined at \(15 ^ { \circ }\) to the plane, as shown in Fig. 6.2. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{5a1895e1-abe3-4739-876a-f19458f0f6ed-5_359_559_1599_830} \captionsetup{labelformat=empty} \caption{Fig. 6.2}
   \end{figure}
3. Draw a diagram showing all the forces acting on the box.
4. Calculate \(P\).
5. Calculate the normal reaction of the plane on the box.

1 Fig. 1.1 shows a circular cylinder of mass 100 kg being raised by a light, inextensible vertical wire AB . There is negligible air resistance. \begin{figure}[h] \end{figure}

1. Calculate the acceleration of the cylinder when the tension in the wire is 1000 N .
2. Calculate the tension in the wire when the cylinder has an upward acceleration of \(0.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). The cylinder is now raised inside a fixed smooth vertical tube that prevents horizontal motion but provides negligible resistance to the upward motion of the cylinder. When the wire is inclined at \(30 ^ { \circ }\) to the vertical, as shown in Fig. 1.2, the cylinder again has an upward acceleration of \(0.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{bf477f61-9f8f-418a-86d8-392bc30323b1-1_308_490_1230_849} \captionsetup{labelformat=empty} \caption{Fig. 1.2}
   \end{figure}
3. Calculate the new tension in the wire.

6 A box of weight 147 N is held by light strings AB and BC . As shown in Fig. 7.1, AB is inclined at \(\alpha\) to the horizontal and is fixed at \(\mathrm { A } ; \mathrm { BC }\) is held at C . The box is in equilibrium with BC horizontal and \(\alpha\) such that \(\sin \alpha = 0.6\) and \(\cos \alpha = 0.8\). \begin{figure}[h] \end{figure}

1. Calculate the tension in string AB .
2. Show that the tension in string BC is 196 N . As shown in Fig. 7.2, a box of weight 90 N is now attached at C and another light string CD is held at D so that the system is in equilibrium with BC still horizontal. CD is inclined at \(\beta\) to the horizontal. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{bf477f61-9f8f-418a-86d8-392bc30323b1-4_378_695_1282_687} \captionsetup{labelformat=empty} \caption{Fig. 7.2}
   \end{figure}
3. Explain why the tension in the string BC is still 196 N .
4. Draw a diagram showing the forces acting on the box at C . Find the angle \(\beta\) and show that the tension in CD is 216 N , correct to three significant figures. The string section CD is now taken over a smooth pulley and attached to a block of mass \(M \mathrm {~kg}\) on a rough slope inclined at \(40 ^ { \circ }\) to the horizontal. As shown in Fig. 7.3, the part of the string attached to the box is still at \(\beta\) to the horizontal and the part attached to the block is parallel to the slope. The system is in equilibrium with a frictional force of 20 N acting on the block up the slope. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{bf477f61-9f8f-418a-86d8-392bc30323b1-5_436_1049_524_536} \captionsetup{labelformat=empty} \caption{Fig. 7.3}
   \end{figure}
5. Calculate the value of \(M\).

7 A block of mass 4 kg is in equilibrium on a rough plane inclined at \(60 ^ { \circ }\) to the horizontal, as shown in Fig. 4. A frictional force of 10 N acts up the plane and a vertical string AB attached to the block is in tension. \begin{figure}[h] \end{figure}

1. Draw a diagram showing the four forces acting on the block.
2. By considering the components of the forces parallel to the slope, calculate the tension in the string.
3. Calculate the normal reaction of the plane on the block.

5 Fig. 5 shows a block of mass 10 kg at rest on a rough horizontal floor. A light string, at an angle of \(30 ^ { \circ }\) to the vertical, is attached to the block. The tension in the string is 50 N . The block is in equilibrium. \begin{figure}[h] \end{figure}

1. Show all the forces acting on the block.
2. Show that the frictional force acting on the block is 25 N .
3. Calculate the normal reaction of the floor on the block.
4. Calculate the magnitude of the total force the floor is exerting on the block.

1. The diagram below shows four coplanar horizontal forces of magnitude \(F \mathrm {~N} , 12 \mathrm {~N} , 16 \mathrm {~N}\) and 20 N acting at a point \(P\) in the directions shown. \includegraphics[max width=\textwidth, alt={}, center]{8f47b2ff-f954-42ec-8ecc-fc64313a7b89-14_792_862_593_607}

Given that the forces are in equilibrium, calculate the value of \(F\) and the size of the angle \(\alpha\). [7]

1. A cyclist is travelling around a circular track which is banked at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac { 3 } { 4 }\)

The cyclist moves with constant speed in a horizontal circle of radius \(r\).
In an initial model,

• the cyclist and her cycle are modelled as a particle
• the track is modelled as being rough so that there is sideways friction between the tyres of the cycle and the track, with coefficient of friction \(\mu\), where \(\mu < \frac { 4 } { 3 }\) Using this model, the maximum speed that the cyclist can travel around the track in a horizontal circle of radius \(r\), without slipping sideways, is \(V\).
  1. Show that \(V = \sqrt { \frac { ( 3 + 4 \mu ) r g } { 4 - 3 \mu } }\)

In a new simplified model,

• the cyclist and her cycle are modelled as a particle
• the motion is now modelled so that there is no sideways friction between the tyres of the cycle and the track

Using this new model, the speed that the cyclist can travel around the track in a horizontal circle of radius \(r\), without slipping sideways, is \(U\).

Find \(U\) in terms of \(r\) and \(g\).

Show that \(U < V\).

1. A girl is cycling round a circular track.

The girl and her bicycle have a combined mass of 55 kg .
The coefficient of friction between the track surface and the tyres of the bicycle is \(\mu\).
The track is banked at an angle of \(15 ^ { \circ }\) to the horizontal.
The girl and her bicycle are modelled as a particle moving in a horizontal circle of radius 50 m
The minimum speed at which the girl can cycle round this circle without slipping is \(4.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) Using the model, find the value of \(\mu\).

8 A rough slope is inclined at an angle of \(25 ^ { \circ }\) to the horizontal. A box of weight 80 newtons is on the slope. A rope is attached to the box and is parallel to the slope. The tension in the rope is of magnitude \(T\) newtons. The diagram shows the slope, the box and the rope. \includegraphics[max width=\textwidth, alt={}, center]{c220e6c4-2676-4022-8301-7d720dc082b2-7_307_469_500_840}

1. The box is held in equilibrium by the rope.
   1. Show that the normal reaction force between the box and the slope is 72.5 newtons, correct to three significant figures.
   2. The coefficient of friction between the box and the slope is 0.32 . Find the magnitude of the maximum value of the frictional force which can act on the box.
   3. Find the least possible tension in the rope to prevent the box from moving down the slope.
   4. Find the greatest possible tension in the rope.
   5. Show that the mass of the box is approximately 8.16 kg .
2. The rope is now released and the box slides down the slope. Find the acceleration of the box.

3 A particle of mass 3 kg is on a smooth slope inclined at \(60 ^ { \circ }\) to the horizontal. The particle is held at rest by a force of \(T\) newtons parallel to the slope, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{fe8c1ea4-cf4d-4741-8af5-03e8c2c88559-2_337_284_2023_879}

1. Draw a diagram to show all the forces acting on the particle.
2. Show that the magnitude of the normal reaction acting on the particle is 14.7 newtons.
3. Find \(T\).

8 A crate, of mass 200 kg , is initially at rest on a rough horizontal surface. A smooth ring is attached to the crate. A light inextensible rope is passed through the ring, and each end of the rope is attached to a tractor. The lower part of the rope is horizontal and the upper part is at an angle of \(20 ^ { \circ }\) to the horizontal, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{fe8c1ea4-cf4d-4741-8af5-03e8c2c88559-5_344_1186_518_420} When the tractor moves forward, the crate accelerates at \(0.3 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). The coefficient of friction between the crate and the surface is 0.4 . Assume that the tension, \(T\) newtons, is the same in both parts of the rope.

1. Draw and label a diagram to show the forces acting on the crate.
2. Express the normal reaction between the surface and the crate in terms of \(T\).
3. Find \(T\).

3 A sign, of mass 2 kg , is suspended from the ceiling of a supermarket by two light strings. It hangs in equilibrium with each string making an angle of \(35 ^ { \circ }\) to the vertical, as shown in the diagram. Model the sign as a particle. \includegraphics[max width=\textwidth, alt={}, center]{81f3753c-f148-44be-8b35-0a8e531016dd-2_424_385_1790_824}

1. By resolving forces horizontally, show that the tension is the same in each string.
2. Find the tension in each string.
3. If the tension in a string exceeds 40 N , the string will break. Find the mass of the heaviest sign that could be suspended as shown in the diagram.

9 Two forces, of magnitudes 2 N and 5 N , act on a particle in the directions shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{ec83c2c5-f8f8-4357-abfa-d40bc1d026b4-07_323_755_548_283}

1. Calculate the magnitude of the resultant force on the particle.
2. Calculate the angle between this resultant force and the force of magnitude 5 N .

7. \begin{figure}[h] \end{figure} A wooden crate of mass 20 kg is pulled in a straight line along a rough horizontal floor using a handle attached to the crate.
The handle is inclined at an angle \(\alpha\) to the floor, as shown in Figure 1, where \(\tan \alpha = \frac { 3 } { 4 }\) The tension in the handle is 40 N .
The coefficient of friction between the crate and the floor is 0.14
The crate is modelled as a particle and the handle is modelled as a light rod.
Using the model,

1. find the acceleration of the crate. The crate is now pushed along the same floor using the handle. The handle is again inclined at the same angle \(\alpha\) to the floor, and the thrust in the handle is 40 N as shown in Figure 2 below. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{65e4b254-fb7b-45c2-9702-32f034018193-20_220_923_1457_571} \captionsetup{labelformat=empty} \caption{Figure 2}
   \end{figure}
2. Explain briefly why the acceleration of the crate would now be less than the acceleration of the crate found in part (a).

7 \includegraphics[max width=\textwidth, alt={}, center]{f0c32e07-f3a0-4d58-bd00-c266177ceaac-3_343_401_1439_872} The diagram shows two forces of magnitudes 10 N and 15 N acting in a horizontal plane on a particle \(P\).

1. Find the component of the 15 N force which is parallel to the 10 N force.
2. Write down the component of the 15 N force which is perpendicular to the 10 N force.
3. Hence, or otherwise, calculate the magnitude and direction of the resultant force on \(P\).

9 A particle of mass \(m \mathrm {~kg}\) rests in equilibrium on a rough horizontal table. There is a string attached to the particle. The tension in the string is \(T \mathrm {~N}\) at an angle of \(\theta\) to the horizontal, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{2e3f056c-58a2-4466-94ea-3fb873e54752-4_205_547_1027_799}

1. Copy and complete the diagram to show all the forces acting on the particle.
2. The coefficient of friction between the particle and the table is \(\mu\) and the particle is on the point of slipping. Show that \(T = \frac { \mu m g } { \cos \theta + \mu \sin \theta }\).
3. Given that \(\mu = 0.75\), find the value of \(\theta\) for which \(T\) is a minimum.

6 \includegraphics[max width=\textwidth, alt={}, center]{9ddae838-2639-4952-bbc0-3944a81e5762-3_401_1224_1315_456} The diagram shows a barge being towed along a canal by a force of 240 N at an angle of \(25 ^ { \circ }\) to its direction of motion. A force, \(F \mathrm {~N}\), perpendicular to the direction of motion, is applied to the barge to keep it moving in the direction shown.

1. Find the magnitude of \(F\).
2. The mass of the barge is 1100 kg and there is a resistance force of 100 N parallel to the direction of motion. Find the acceleration of the barge.

11 \includegraphics[max width=\textwidth, alt={}, center]{35d24778-1203-4d5d-be4b-bb375344fe09-4_285_700_1043_721} Three forces are acting on a particle \(A\) as shown in the diagram. The forces act in the same plane and the particle is in equilibrium.

1. Evaluate \(P\) and \(\theta\). The 8 N force is removed.
2. State the direction of the instantaneous acceleration of \(A\).

\includegraphics{figure_2} Coplanar forces of magnitudes 20 N, \(P\) N, \(3P\) N and \(4P\) N act at a point in the directions shown in the diagram. The system is in equilibrium. Find \(P\) and \(\theta\). [6]

\includegraphics{figure_3} A particle of mass 2.5 kg is held in equilibrium on a rough plane inclined at 20° to the horizontal by a force of magnitude \(T\) N making an angle of 60° with a line of greatest slope of the plane (see diagram). The coefficient of friction between the particle and the plane is 0.3. Find the greatest and least possible values of \(T\). [8]

\includegraphics{figure_6} Three coplanar forces of magnitudes 10 N, 25 N and 20 N act at a point \(O\) in the directions shown in the diagram.

1. Given that the component of the resultant force in the \(x\)-direction is zero, find \(\alpha\), and hence find the magnitude of the resultant force. [4]
2. Given instead that \(\alpha = 45\), find the magnitude and direction of the resultant of the three forces. [5]

A crate of mass 300 kg is at rest on rough horizontal ground. The coefficient of friction between the crate and the ground is 0.5. A force of magnitude \(X\) N, acting at an angle \(\alpha\) above the horizontal, is applied to the crate, where \(\sin \alpha = 0.28\). Find the greatest value of \(X\) for which the crate remains at rest. [5]

\includegraphics{figure_4} Three coplanar forces of magnitudes 20 N, 100 N and \(F\) N act at a point. The directions of these forces are shown in the diagram. Given that the three forces are in equilibrium, find \(F\) and \(\alpha\). [6]

\includegraphics{figure_5} A block of mass \(12\text{kg}\) is placed on a plane which is inclined at an angle of \(24°\) to the horizontal. A light string, making an angle of \(36°\) above a line of greatest slope, is attached to the block. The tension in the string is \(65\text{N}\) (see diagram). The coefficient of friction between the block and plane is \(\mu\). The block is in limiting equilibrium and is on the point of sliding up the plane. Find \(\mu\). [6]