3.03a Force: vector nature and diagrams

2. A particle \(P\) of mass 0.75 kg is moving under the action of a single force \(\mathbf { F }\) newtons. At time \(t\) seconds, the velocity \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\) of \(P\) is given by $$\mathbf { v } = \left( t ^ { 2 } + 2 \right) \mathbf { i } - 6 t \mathbf { j }$$

1. Find the magnitude of \(\mathbf { F }\) when \(t = 4\).
   (5) When \(t = 5\), the particle \(P\) receives an impulse of magnitude \(9 \sqrt { } 2 \mathrm { Ns }\) in the direction of the vector \(\mathbf { i } - \mathbf { j }\).
2. Find the velocity of \(P\) immediately after the impulse.

4. A particle \(P\) of mass 0.4 kg is moving under the action of a single force \(\mathbf { F }\) newtons. At time \(t\) seconds, the velocity of \(P , \mathbf { v } \mathrm {~m} \mathrm {~s} ^ { - 1 }\), is given by $$\mathbf { v } = ( 6 t + 4 ) \mathbf { i } + \left( t ^ { 2 } + 3 t \right) \mathbf { j } .$$ When \(t = 0 , P\) is at the point with position vector \(( - 3 \mathbf { i } + 4 \mathbf { j } ) \mathrm { m }\). When \(t = 4 , P\) is at the point \(S\).

1. Calculate the magnitude of \(\mathbf { F }\) when \(t = 4\).
2. Calculate the distance \(O S\).

5. A car of mass 1000 kg is towing a trailer of mass 1500 kg along a straight horizontal road. The tow-bar joining the car to the trailer is modelled as a light rod parallel to the road. The total resistance to motion of the car is modelled as having constant magnitude 750 N . The total resistance to motion of the trailer is modelled as of magnitude \(R\) newtons, where \(R\) is a constant. When the engine of the car is working at a rate of 50 kW , the car and the trailer travel at a constant speed of \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Show that \(R = 1250\). When travelling at \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) the driver of the car disengages the engine and applies the brakes. The brakes provide a constant braking force of magnitude 1500 N to the car. The resisting forces of magnitude 750 N and 1250 N are assumed to remain unchanged. Calculate
2. the deceleration of the car while braking,
3. the thrust in the tow-bar while braking,
4. the work done, in kJ , by the braking force in bringing the car and the trailer to rest.
5. Suggest how the modelling assumption that the resistances to motion are constant could be refined to be more realistic.

2. A spacecraft \(S\) of mass \(m\) moves in a straight line towards the centre of the Earth. The Earth is modelled as a sphere of radius \(R\) and \(S\) is modelled as a particle. When \(S\) is at a distance \(x , x \geqslant R\), from the centre of the Earth, the force exerted by the Earth on \(S\) is directed towards the centre of the Earth. The force has magnitude \(\frac { K } { x ^ { 2 } }\), where \(K\) is a constant.

1. Show that \(K = m g R ^ { 2 }\) When \(S\) is at a distance \(3 R\) from the centre of the Earth, the speed of \(S\) is \(V\). Assuming that air resistance can be ignored,
2. find, in terms of \(g , R\) and \(V\), the speed of \(S\) as it hits the surface of the Earth.

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2. \begin{figure}[h] \end{figure} Figure 1 shows a regular hexagon \(O P Q R S T\).
The vectors \(\mathbf { p }\) and \(\mathbf { q }\) are defined by \(\mathbf { p } = \overrightarrow { O P }\) and \(\mathbf { q } = \overrightarrow { O Q }\) Forces, in Newtons, \(\mathbf { F } _ { P } = ( \overrightarrow { O P } ) , \mathbf { F } _ { Q } = 2 \times ( \overrightarrow { O Q } ) , \mathbf { F } _ { R } = 3 \times ( \overrightarrow { O R } ) , \mathbf { F } _ { S } = 4 \times ( \overrightarrow { O S } )\) and \(\mathbf { F } _ { T } = 5 \times ( \overrightarrow { O T } )\) are applied to a particle.

1. Find, in terms of \(\mathbf { p }\) and \(\mathbf { q }\), the resultant force on the particle. The magnitude of the acceleration of the particle due to these forces is \(13 \mathrm {~ms} ^ { - 2 }\) Given that the mass of the particle is 3 kg ,
2. find \(| \mathbf { p } |\) \includegraphics[max width=\textwidth, alt={}, center]{71cd126f-1c7d-4e37-a26d-7ff98a74fd79-04_2255_56_310_1980}

3 Two forces of magnitudes 8 N and 12 N act at a point \(O\).

1. Given that the two forces are perpendicular to each other, find
   1. the angle between the resultant and the 12 N force,
   2. the magnitude of the resultant.
   3. It is given instead that the resultant of the two forces has magnitude \(R \mathrm {~N}\) and acts in a direction perpendicular to the 8 N force (see diagram). \includegraphics[max width=\textwidth, alt={}, center]{b7f05d10-9d3c-4098-846d-ca6511c75c5d-2_248_388_1877_826}
      (a) Calculate the angle between the resultant and the 12 N force.
      (b) Find \(R\).

4 \includegraphics[max width=\textwidth, alt={}, center]{8b79facc-e37f-45c3-95c0-9f2a30ca8fe4-3_394_963_276_552} Two forces of magnitudes 6 N and 10 N separated by an angle of \(110 ^ { \circ }\) act on a particle \(P\), which rests on a horizontal surface (see diagram).

1. Find the magnitude of the resultant of the 6 N and 10 N forces, and the angle between the resultant and the 10 N force. The two forces act in the same vertical plane. The particle \(P\) has weight 20 N and rests in equilibrium on the surface. Given that the surface is smooth, find
2. the magnitude of the force exerted on \(P\) by the surface,
3. the angle between the surface and the 10 N force.

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 Fig. 1 shows a pile of four uniform blocks in equilibrium on a horizontal table. Their masses, as shown, are \(4 \mathrm {~kg} , 5 \mathrm {~kg} , 7 \mathrm {~kg}\) and 10 kg . \begin{figure}[h] \end{figure} Mark on the diagram the magnitude and direction of each of the forces acting on the 7 kg block.

9 In this question the horizontal unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are in the directions east and north respectively.
A model ship of mass 2 kg is moving so that its acceleration vector \(\mathbf { a m s } ^ { - 2 }\) at time \(t\) seconds is given by \(\mathbf { a } = 3 ( 2 t - 5 ) \mathbf { i } + 4 \mathbf { j }\). When \(t = T\), the magnitude of the horizontal force acting on the ship is 10 N . Find the possible values of \(T\).

9 A crane lifts a car vertically. The car is inside a crate which is raised by the crane by means of a strong cable. The cable can withstand a maximum tension of 9500 N without breaking. The crate has a mass of 55 kg and the car has a mass of 830 kg .

1. Find the maximum acceleration with which the crate and car can be raised.
2. Show on a clearly labelled diagram the forces acting on the crate while it is in motion.
3. Determine the magnitude of the reaction force between the crate and the car when they are ascending with maximum acceleration.

10 A rescue worker is lowered from a helicopter on a rope. She attaches a second rope to herself and to a woman in difficulties on the ground. The helicopter winches both women upwards with the rescued woman vertically below the rescue worker, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{5428eabf-431d-4db1-8c25-1f2b9570d9aa-6_509_460_408_262} The model for this motion uses the following modelling assumptions:

• each woman can be modelled as a particle;
• the ropes are both light and inextensible;
• there is no air resistance to the motion;
• the motion is in a vertical line.
  1. Explain what it means when the women are each 'modelled as a particle'.
  2. Explain what 'light' means in this context.

The tension in the rope to the helicopter is 1500 N . The rescue worker has a mass of 65 kg and the rescued woman has a mass of 75 kg .

Draw a diagram showing the forces on the two women.

Write down the equation of motion of the two women considered as a single particle.

Calculate the acceleration of the women.

Determine the tension in the rope connecting the two women.

13 A 15 kg box is suspended in the air by a rope which makes an angle of \(30 ^ { \circ }\) with the vertical. The box is held in place by a string which is horizontal.

1. Draw a diagram showing the forces acting on the box.
2. Calculate the tension in the rope.
3. Calculate the tension in the string.

11 A block of mass 2 kg is placed on a rough horizontal table. A light inextensible string attached to the block passes over a smooth pulley attached to the edge of the table. The other end of the string is attached to a sphere of mass 0.8 kg which hangs freely. The part of the string between the block and the pulley is horizontal. The coefficient of friction between the table and the block is 0.35 . The system is released from rest.

1. Draw a force diagram showing all the forces on the block and the sphere.
2. Write down the equations of motion for the block and the sphere.
3. Show that the acceleration of the system is \(0.35 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
4. Calculate the time for the block to slide the first 0.5 m . Assume the block does not reach the pulley.

9 The diagram shows a toy caterpillar consisting of a head and three body sections each connected by a light inextensible ribbon. The head has a mass of 120 g and the body sections each have a mass of 90 g . The toy is pulled on level ground using a horizontal string attached to the head. The tension in the string is 12 N . There are resistances to motion of 2.5 N for the head and each section of the body. \includegraphics[max width=\textwidth, alt={}, center]{4fac72cb-85cb-48d9-8817-899ef3f80a0f-08_134_794_536_244}

1. 1. State the equation of motion for the toy caterpillar modelled as a single particle.
   2. Calculate the acceleration of the toy caterpillar.
2. Draw a diagram showing all the forces acting on the head of the toy caterpillar.
3. Calculate the tension in the ribbon that joins the head to the body.

5 A woman is pulling a loaded sledge along horizontal ground. The only resistance to motion of the sledge is due to friction between it and the ground. \begin{figure}[h] \end{figure} At first, she pulls with a force of 100 N inclined at \(32 ^ { \circ }\) to the horizontal, as shown in Fig.5, but the sledge does not move.

1. Determine the frictional force between the ground and the sledge. Give your answer correct to 3 significant figures.
2. Next she pulls with a force of 100 N inclined at a smaller angle to the horizontal. The sledge still does not move. Compare the frictional force in this new situation with that in part (a), justifying your answer.

2 The diagram shows three forces and the perpendicular unit vectors \(\mathbf { i }\) and \(\mathbf { j }\), which all lie in the same plane. \includegraphics[max width=\textwidth, alt={}, center]{a381686b-0b1e-41ba-b88f-be1601e42098-2_415_398_1507_605} \includegraphics[max width=\textwidth, alt={}, center]{a381686b-0b1e-41ba-b88f-be1601e42098-2_172_166_1567_1217}

1. Express the resultant of the three forces in terms of \(\mathbf { i }\) and \(\mathbf { j }\).
2. Find the magnitude of the resultant force.
3. Draw a diagram to show the direction of the resultant force, and find the angle that it makes with the unit vector \(\mathbf { i }\).

3. A particle is in equilibrium under the action of three forces \(\mathbf { P } , \mathbf { Q }\) and \(\mathbf { R }\) acting in the same horizontal plane. \(P\) has magnitude 9 N and acts on a bearing of \(030 ^ { \circ } . Q\) has magnitude 12 N . and acts on a bearing of \(225 ^ { \circ }\).

1. Find the values of \(a\) and \(b\) such that \(\mathbf { R } = ( a \mathbf { i } + b \mathbf { j } ) \mathrm { N }\), where \(\mathbf { i }\) and \(\mathbf { j }\) are unit vectors in the directions due East and due North respectively.
2. Calculate the magnitude and direction of \(\mathbf { R }\)

1 Fig. 2 shows a 6 kg block on a smooth horizontal table. It is connected to blocks of mass 2 kg and 9 kg by two light strings which pass over smooth pulleys at the edges of the table. The parts of the strings attached to the 6 kg block are horizontal. \begin{figure}[h] \end{figure}

1. Draw three separate diagrams showing all the forces acting on each of the blocks.
2. Calculate the acceleration of the system and the tension in each string.

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.

3 Fig. 2 shows a sack of rice of weight 250 N hanging in equilibrium supported by a light rope AB . End A of the rope is attached to the sack. The rope passes over a small smooth fixed pulley. \begin{figure}[h] \end{figure} Initially, end B of the rope is attached to a vertical wall as shown in Fig. 2.

1. Calculate the horizontal and the vertical forces acting on the wall due to the rope. End B of the rope is now detached from the wall and attached instead to the top of the sack. The sack is in equilibrium with both sections of the rope vertical.
2. Calculate the tension in the rope.

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.

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.