3.03e Resolve forces: two dimensions

3. \begin{figure}[h] \end{figure} A particle \(P\) of mass \(m\) is held at rest at a point on a rough inclined plane, as shown in Figure 3. It is given that

• the plane is inclined to the horizontal at an angle \(\alpha\), where \(\tan \alpha = \frac { 5 } { 12 }\)
• the coefficient of friction between \(P\) and the plane is \(\mu\), where \(\mu < \frac { 5 } { 12 }\)

The particle \(P\) is released from rest and slides down the plane.
Air resistance is modelled as being negligible.
Using the model,

1. find, in terms of \(m\) and \(g\), the magnitude of the normal reaction of the plane on \(P\),
2. show that, as \(P\) slides down the plane, the acceleration of \(P\) down the plane is $$\frac { 1 } { 13 } g ( 5 - 12 \mu )$$
3. State what would happen to \(P\) if it is released from rest but \(\mu \geqslant \frac { 5 } { 12 }\)

3 The diagram shows a rope that is attached to a box of mass 25 kg , which is being pulled along rough horizontal ground. The rope is at an angle of \(30 ^ { \circ }\) to the ground. The tension in the rope is 40 N . The box accelerates at \(0.1 \mathrm {~ms} ^ { - 2 }\). \includegraphics[max width=\textwidth, alt={}, center]{eb1f2470-aeeb-4b1d-a6c0-bdeb7048edd5-3_214_729_504_644}

1. Draw a diagram to show all of the forces acting on the box.
2. Show that the magnitude of the friction force acting on the box is 32.1 N , correct to three significant figures.
3. Show that the magnitude of the normal reaction force that the ground exerts on the box is 225 N .
4. Find the coefficient of friction between the box and the ground.
5. State what would happen to the magnitude of the friction force if the angle between the rope and the horizontal were increased. Give a reason for your answer.

3 The diagram shows three forces which act in the same plane and are in equilibrium. \includegraphics[max width=\textwidth, alt={}, center]{965a176a-848c-478d-a748-80fc9dfe4399-3_419_516_383_761}

1. Find \(F\).
2. Find \(\alpha\).

6 A trolley, of mass 100 kg , rolls at a constant speed along a straight line down a slope inclined at an angle of \(4 ^ { \circ }\) to the horizontal. Assume that a constant resistance force, of magnitude \(P\) newtons, acts on the trolley as it moves. Model the trolley as a particle.

1. Draw a diagram to show the forces acting on the trolley.
2. Show that \(P = 68.4 \mathrm {~N}\), correct to three significant figures.
3. 1. Find the acceleration of the trolley if it rolls down a slope inclined at \(5 ^ { \circ }\) to the horizontal and experiences the same constant force of magnitude \(P\) that you found in part (b).
   2. Make one criticism of the assumption that the resistance force on the trolley is constant.

3 A particle, of mass 4 kg , is suspended in equilibrium by two light strings, \(A P\) and \(B P\). The string \(A P\) makes an angle of \(30 ^ { \circ }\) to the horizontal and the other string, \(B P\), is horizontal, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{217f0e3e-9d1b-41f1-8299-f56d073fbbeb-2_231_757_1841_639}

1. Draw and label a diagram to show the forces acting on the particle.
2. Show that the tension in the string \(A P\) is 78.4 N .
3. Find the tension in the horizontal string \(B P\).

5 A sledge of mass 8 kg is at rest on a rough horizontal surface. A child tries to move the sledge by pushing it with a pole, as shown in the diagram, but the sledge does not move. The pole is at an angle of \(30 ^ { \circ }\) to the horizontal and exerts a force of 40 newtons on the sledge. \includegraphics[max width=\textwidth, alt={}, center]{8c6f9ac0-c24f-48d0-9fb2-883651e791d7-4_221_922_513_552} Model the sledge as a particle.

1. Draw a diagram to show the four forces acting on the sledge.
2. Show that the normal reaction force between the sledge and the surface has magnitude 98.4 N .
3. Find the magnitude of the friction force that acts on the sledge.
4. Find the least possible value of the coefficient of friction between the sledge and the surface.

8 A van, of mass 2000 kg , is towed up a slope inclined at \(5 ^ { \circ }\) to the horizontal. The tow rope is at an angle of \(12 ^ { \circ }\) to the slope. The motion of the van is opposed by a resistance force of magnitude 500 newtons. The van is accelerating up the slope at \(0.6 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). \includegraphics[max width=\textwidth, alt={}, center]{f30b02da-a41e-44cb-b45f-9e6a3a9d0528-22_269_991_513_529} Model the van as a particle.

1. Draw a diagram to show the forces acting on the van.
2. Show that the tension in the tow rope is 3480 newtons, correct to three significant figures.

3 A box, of mass 3 kg , is placed on a rough slope inclined at an angle of \(40 ^ { \circ }\) to the horizontal. It is released from rest and slides down the slope.

1. Draw a diagram to show the forces acting on the box.
2. Find the magnitude of the normal reaction force acting on the box.
3. The coefficient of friction between the box and the slope is 0.2 . Find the magnitude of the friction force acting on the box.
4. Find the acceleration of the box.
5. State an assumption that you have made about the forces acting on the box.

8 A rough slope is inclined at an angle of \(10 ^ { \circ }\) to the horizontal. A particle of mass 6 kg is on the slope. A string is attached to the particle and is at an angle of \(30 ^ { \circ }\) to the slope. The tension in the string is 20 N . The diagram shows the slope, the particle and the string. \includegraphics[max width=\textwidth, alt={}, center]{7e0585ea-062a-487c-8e39-37a4ed414ff8-6_259_684_518_676} The particle moves up the slope with an acceleration of \(0.4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

1. Draw a diagram to show the forces acting on the particle.
2. Show that the magnitude of the normal reaction force is 47.9 N , correct to three significant figures.
3. Find the coefficient of friction between the particle and the slope.

4 A block is being pulled up a rough plane inclined at an angle of \(22 ^ { \circ }\) to the horizontal by a rope parallel to the plane, as shown in the diagram. The mass of the block is 0.7 kg , and the tension in the rope is \(T\) newtons. \includegraphics[max width=\textwidth, alt={}, center]{cfe0bdbc-35e3-485f-a922-b652a72f4c95-3_264_460_1649_779}

1. Draw a diagram to show the forces acting on the block.
2. Show that the normal reaction force between the block and the plane has magnitude 6.36 newtons, correct to three significant figures.
3. The coefficient of friction between the block and the plane is 0.25 . Find the magnitude of the frictional force acting on the block during its motion.
4. The tension in the rope is 5.6 newtons. Find the acceleration of the block.

6 A block, of mass 5 kg , slides down a rough plane inclined at \(40 ^ { \circ }\) to the horizontal. When modelling the motion of the block, assume that there is no air resistance acting on it.

1. Draw and label a diagram to show the forces acting on the block.
2. Show that the magnitude of the normal reaction force acting on the block is 37.5 N , correct to three significant figures.
3. Given that the acceleration of the block is \(0.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), find the coefficient of friction between the block and the plane.
4. In reality, air resistance does act on the block. State how this would change your value for the coefficient of friction and explain why.

8 The diagram shows a block, of mass 20 kg , being pulled along a rough horizontal surface by a rope inclined at an angle of \(30 ^ { \circ }\) to the horizontal. \includegraphics[max width=\textwidth, alt={}, center]{c022c936-72bc-4cf9-8f98-285f12c1d479-16_323_1194_411_424} The coefficient of friction between the block and the surface is \(\mu\). Model the block as a particle which slides on the surface.

1. If the tension in the rope is 60 newtons, the block moves at a constant speed.
   1. Show that the magnitude of the normal reaction force acting on the block is 166 N .
   2. Find \(\mu\).
2. If the rope remains at the same angle and the block accelerates at \(0.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), find the tension in the rope. \includegraphics[max width=\textwidth, alt={}, center]{c022c936-72bc-4cf9-8f98-285f12c1d479-18_2488_1719_219_150} \includegraphics[max width=\textwidth, alt={}, center]{c022c936-72bc-4cf9-8f98-285f12c1d479-19_2488_1719_219_150}

4 A particle, of mass \(m \mathrm {~kg}\), remains in equilibrium under the action of three forces, which act in a vertical plane, as shown in the diagram. The force with magnitude 60 N acts at \(48 ^ { \circ }\) above the horizontal and the force with magnitude 50 N acts at an angle \(\theta\) above the horizontal. \includegraphics[max width=\textwidth, alt={}, center]{5d474771-fe32-47c6-8bf3-60ff7a25dd12-08_576_647_548_701}

1. By resolving horizontally, find \(\theta\).
2. Find \(m\).
   \includegraphics[max width=\textwidth, alt={}]{5d474771-fe32-47c6-8bf3-60ff7a25dd12-09_2484_1709_223_153}

8 Three forces act in a vertical plane on an object of mass 250 kg , as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{7ac7dfd0-4c3e-4eb7-920f-ce5b24ad1281-5_481_1139_408_447} The two forces \(P\) newtons and \(Q\) newtons each act at \(80 ^ { \circ }\) to the horizontal. The object accelerates horizontally at \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\) under the action of these forces.

1. Show that $$P = 125 \left( \frac { a } { \cos 80 ^ { \circ } } + \frac { g } { \sin 80 ^ { \circ } } \right)$$
2. Find the value of \(a\) for which \(Q\) is zero.

3 A ship has a mass of 500 tonnes. Two tugs are used to pull the ship using cables that are horizontal. One tug exerts a force of 100000 N at an angle of \(25 ^ { \circ }\) to the centre line of the ship. The other tug exerts a force of \(T \mathrm {~N}\) at an angle of \(20 ^ { \circ }\) to the centre line of the ship. The diagram shows the ship and forces as viewed from above. \includegraphics[max width=\textwidth, alt={}, center]{01338c87-302c-420f-a473-39b0796ccaed-06_279_844_539_664} The ship accelerates in a straight line along its centre line.

1. \(\quad\) Find \(T\).
2. A resistance force of magnitude 20000 N directly opposes the motion of the ship. Find the acceleration of the ship.
   [0pt] [4 marks]
   \includegraphics[max width=\textwidth, alt={}]{01338c87-302c-420f-a473-39b0796ccaed-06_1419_1714_1288_153}

7 Two forces, which act in a vertical plane, are applied to a crate. The crate has mass 50 kg , and is initially at rest on a rough horizontal surface. One force has magnitude 80 N and acts at an angle of \(30 ^ { \circ }\) to the horizontal and the other has magnitude 40 N and acts at an angle of \(20 ^ { \circ }\) to the horizontal. The forces are shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{01338c87-302c-420f-a473-39b0796ccaed-16_241_999_493_523} The coefficient of friction between the crate and the surface is 0.6 . Model the crate as a particle.

1. Draw a diagram to show the forces acting on the crate.
2. Find the magnitude of the normal reaction force acting on the crate.
3. Does the crate start to move when the two forces are applied to the crate? Show all your working.
4. State one aspect of the possible motion of the crate that is ignored by modelling it as a particle.
   [0pt] [1 mark]

6 A floor polisher consists of a heavy metal block with a polishing cloth attached to the underside. A light rod is also attached to the block and is used to push the block across the floor that is to be polished. The block has mass 5 kg . Assume that the floor is horizontal. Model the block as a particle. The coefficient of friction between the cloth and the floor is 0.2 .
A person pushes the rod to exert a force on the block. The force is at an angle of \(60 ^ { \circ }\) to the horizontal and the block accelerates at \(0.9 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). The diagram shows the block and the force exerted by the rod in this situation. \includegraphics[max width=\textwidth, alt={}, center]{5dd17095-18a6-470b-a24a-4456c8e3ed31-14_309_205_772_1009} The rod exerts a force of magnitude \(T\) newtons on the block.

1. Find, in terms of \(T\), the magnitude of the normal reaction force acting on the block.
2. \(\quad\) Find \(T\).
   [0pt] [6 marks]

3. \begin{figure}[h] \end{figure} Figure 2 shows a ball of mass 3 kg lying on a smooth plane inclined at an angle \(\alpha\) to the horizontal where \(\sin \alpha = \frac { 3 } { 5 }\). The ball is held in equilibrium by a force of magnitude \(P\) newtons, which acts at an angle of \(10 ^ { \circ }\) to the line of greatest slope of the plane.

1. Suggest a suitable model for the ball. Giving your answers correct to 1 decimal place,
2. find the value of \(P\),
3. find the magnitude of the reaction between the ball and the plane.

1. \begin{figure}[h] \end{figure} Figure 1 shows a light, inextensible string fixed at one end to a point \(P\). The other end is attached to a small object of weight 10 N . The object is subjected to a horizontal force \(H\) so that the string makes an angle of \(30 ^ { \circ }\) with the vertical.

1. Find the magnitude of the tension in the string.
2. Show that the ratio of the magnitude of the tension to the magnitude of \(H\) is \(2 : 1\).

7. \begin{figure}[h] \end{figure} Figure 3 shows a particle of mass 4 kg resting on the surface of a rough plane inclined at an angle of \(30 ^ { \circ }\) to the horizontal. It is connected by a light inextensible string passing over a smooth pulley at the top of the plane, to a particle of mass 5 kg which hangs freely. The coefficient of friction between the 4 kg mass and the plane is \(\mu\) and when the system is released from rest the 4 kg mass starts to move up the slope.

1. Show that the acceleration of the system is \(\frac { 1 } { 9 } ( 3 - 2 \mu \sqrt { 3 } ) \mathrm { g } \mathrm { ms } ^ { - 2 }\).
2. Hence, find the maximum value of \(\mu\). Given that \(\mu = \frac { 1 } { 2 }\),
3. find the tension in the string in terms of \(g\),
4. show that the magnitude of the force on the pulley is given by \(\frac { 5 } { 3 } ( 2 \sqrt { 3 } + 1 ) \mathrm { g }\). END

2. \begin{figure}[h] \end{figure} Figure 1 shows a toy lorry being pulled by a piece of string, up a ramp inclined at an angle of \(25 ^ { \circ }\) to the horizontal. When the string is pulled with a force of 20 N parallel to the line of greatest slope of the ramp, the lorry is on the point of moving up the ramp. In a simple model of the situation, the ramp is considered to be smooth.

1. Draw a diagram showing all the forces acting on the lorry.
2. Find the weight of the lorry and the magnitude of the reaction between the lorry and the ramp, giving your answers to an appropriate degree of accuracy.
3. Write down any modelling assumptions that you have made about
   1. the lorry,
   2. the string. In a more refined model, the ramp is assumed to be rough.
4. State the effect that this would have on your answers to part (b).

2. A monk uses a small brush to clean the stone floor of a monastery by pushing the brush with a force of \(P\) Newtons at an angle of \(60 ^ { \circ }\) to the vertical. He moves the brush at a constant speed. The mass of the brush is 0.5 kg and the coefficient of friction between the brush and the floor is \(\frac { 1 } { \sqrt { 3 } }\). The brush is modelled as a particle and air resistance is ignored.

1. Show that \(P = \frac { g } { 2 }\) Newtons.
2. Explain why it is reasonable to ignore air resistance in this situation.

2 Fig. 3 shows two people, Sam and Tom, pushing a car of mass 1000 kg along a straight line \(l\) on level ground. Sam pushes with a constant horizontal force of 300 N at an angle of \(30 ^ { \circ }\) to the line \(l\).
Tom pushes with a constant horizontal force of 175 N at an angle of \(15 ^ { \circ }\) to the line \(l\). \begin{figure}[h] \end{figure}

1. The car starts at rest and moves with constant acceleration. After 6 seconds it has travelled 7.2 m . Find its acceleration.
2. Find the resistance force acting on the car along the line \(l\).
3. The resultant of the forces exerted by Sam and Tom is not in the direction of the car's acceleration. Explain briefly why.

3 A trolley C of mass 8 kg with rusty axle bearings is initially at rest on a horizontal floor.
The trolley stays at rest when it is pulled by a horizontal string with tension 25 N , as shown in Fig. 8.1. \begin{figure}[h] \end{figure}

1. State the magnitude of the horizontal resistance opposing the pull. A second trolley D of mass 10 kg is connected to trolley C by means of a light, horizontal rod.
   The string now has tension 50 N , and is at an angle of \(25 ^ { \circ }\) to the horizontal, as shown in Fig. 8.2. The two trolleys stay at rest. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{f5f9b9b7-6766-4f8e-b011-506051104123-3_297_1180_971_741} \captionsetup{labelformat=empty} \caption{Fig. 8.2}
   \end{figure}
2. Calculate the magnitude of the total horizontal resistance acting on the two trolleys opposing the pull.
3. Calculate the normal reaction of the floor on trolley C . The axle bearings of the trolleys are oiled and the total horizontal resistance to the motion of the two trolleys is now 20 N . The two trolleys are still pulled by the string with tension 50 N , as shown in Fig. 8.2.
4. Calculate the acceleration of the trolleys. In a new situation, the trolleys are on a slope at \(5 ^ { \circ }\) to the horizontal and are initially travelling down the slope at \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The resistances are 15 N to the motion of D and 5 N to the motion of C . There is no string attached. The rod connecting the trolleys is parallel to the slope. This situation is shown in Fig. 8.3. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{f5f9b9b7-6766-4f8e-b011-506051104123-3_351_1285_2038_466} \captionsetup{labelformat=empty} \caption{Fig. 8.3}
   \end{figure}
5. Calculate the speed of the trolleys after 2 seconds and also the force in the rod connecting the PhysicsAptMaths, statter \&REther this rod is in tension or thrust (compression).

1 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.