3.03v Motion on rough surface: including inclined planes

4 Fig. 4 shows a smooth plane inclined at an angle of \(30 ^ { \circ }\) to the horizontal. Two fixed points A and B on the plane are 4.55 m apart with B higher than A on a line of greatest slope. A particle P of mass 0.25 kg is in contact with the plane and is connected to A and to B by two light elastic strings. The string AP has natural length 1.5 m and modulus of elasticity 7.35 N ; the string BP has natural length 2.5 m and modulus of elasticity 7.35 N . The particle P moves along part of the line AB , with both strings taut throughout the motion. \begin{figure}[h] \end{figure}

1. Show that, when \(\mathrm { AP } = 1.55 \mathrm {~m}\), the acceleration of P is zero.
2. Taking \(\mathrm { AP } = ( 1.55 + x ) \mathrm { m }\), write down the tension in the string AP , in terms of \(x\), and show that the tension in the string BP is \(( 1.47 - 2.94 x ) \mathrm { N }\).
3. Show that the motion of P is simple harmonic, and find its period. The particle P is released from rest with \(\mathrm { AP } = 1.5 \mathrm {~m}\).
4. Find the time after release when P is first moving down the plane with speed \(0.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

6. A car is travelling on a horizontal racetrack round a circular bend of radius 40 m . The coefficient of friction between the car and the road is \(\frac { 2 } { 5 }\).

1. Find the maximum speed at which the car can travel round the bend without slipping, giving your answer correct to 3 significant figures.
   (5 marks)
   The owner of the track decides to bank the corner at an angle of \(25 ^ { \circ }\) in order to enable the cars to travel more quickly.
2. Show that this increases the maximum speed at which the car can travel round the bend without slipping by 63\%, correct to the nearest whole number.
   (8 marks)

7. A cyclist is travelling round a circular bend of radius 25 m on a track which is banked at an angle of \(35 ^ { \circ }\) to the horizontal. In a model of the situation, the cyclist and her bicycle are represented by a particle of mass 60 kg and air resistance and friction are ignored. Using this model and assuming that the cyclist is not slipping,

1. find, correct to 3 significant figures, the speed at which she is travelling. In tests it is found that the cyclist must travel at a minimum speed of \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to prevent the bicycle from slipping down the slope. A more refined model is now used with a coefficient of friction between the bicycle and the track of \(\mu\). Using this model,
2. show that \(\mu = 0.227\), correct to 3 significant figures,
3. find, correct to 2 significant figures, the maximum speed at which the cyclist can travel without slipping up the slope. END

4. \begin{figure}[h] \end{figure} A particle \(P\) of mass 2 kg is attached to one end of a light elastic string of natural length 1.5 m and modulus of elasticity \(\lambda\). The other end of the string is fixed to a point \(A\) on a rough plane inclined at an angle of \(30 ^ { \circ }\) to the horizontal as shown in Figure 2. The coefficient of friction between \(P\) and the plane is \(\frac { 1 } { 6 } \sqrt { 3 }\). \(P\) is held at rest at \(A\) and then released. It first comes to instantaneous rest at the point \(B , 2.2 \mathrm {~m}\) from \(A\). For the motion of \(P\) from \(A\) to \(B\),

1. show that the work done against friction is 10.78 J ,
2. find the change in the gravitational potential energy of \(P\). By using the work-energy principle, or otherwise,
3. find \(\lambda\).

3 A box weighing 130 N is on a rough plane inclined at \(12 ^ { \circ }\) to the horizontal.
The box is held at rest on the plane by the action of a force of magnitude 70 N acting up the plane in a direction parallel to a line of greatest slope of the plane.
The box is on the point of slipping up the plane.

1. Find the coefficient of friction between the box and the plane. The force of magnitude 70 N is removed.
2. Determine whether or not the box remains in equilibrium.

6 A block B of mass \(m \mathrm {~kg}\) rests on a rough slope inclined at angle \(\alpha\) to the horizontal. The coefficient of friction between \(B\) and the slope is \(\frac { 5 } { 9 }\).

1. When B is in limiting equilibrium, show that \(\tan \alpha = \frac { 5 } { 9 }\).
2. If \(\alpha = 40 ^ { \circ }\), determine the acceleration of B down the slope. A horizontal force of magnitude \(P \mathrm {~N}\) is now applied to B , as shown in the diagram below. At first B is at rest. \includegraphics[max width=\textwidth, alt={}, center]{d1ec7861-dc8b-450b-8e05-c70479ab0dc2-7_381_410_689_242} \(P\) is gradually increased.
3. Show that, for B to slide on the slope, $$\mathrm { P } \left( \cos \alpha - \frac { 5 } { 9 } \sin \alpha \right) > \mathrm { mg } \left( \frac { 5 } { 9 } \cos \alpha + \sin \alpha \right) .$$
4. Determine, in degrees, the least value of \(\alpha\) for which B will not slide no matter how large \(P\) becomes.

6 Fig. 6.1 shows a cross-section through a block of mass 5 kg which is on top of a trolley of mass 11 kg . The trolley is on top of a smooth horizontal surface. The coefficient of friction between the block and the trolley is 0.3 . Throughout this question you may assume that there are no other resistances to motion on either the block or the trolley. \begin{figure}[h] \end{figure} Initially, both the block and trolley are at rest. A constant force of magnitude 50 N is now applied horizontally to the trolley, as shown in Fig. 6.1.

1. Show that in the subsequent motion the block will slide.
2. Find the acceleration of
   1. the block,
   2. the trolley. The same block and trolley are again at rest. An obstruction, in the form of a fixed horizontal pole, is placed in front of the block, the pole is 91 cm above the trolley and the width of the block is 56 cm as shown in Fig. 6.2, as well as the forward direction of motion. \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{5c1cfe41-d7a2-4f69-ae79-67d9f023c246-6_426_1324_1793_269} \captionsetup{labelformat=empty} \caption{Fig. 6.2}
      \end{figure} It is given that the block is uniform and that the contact between the pole and the block is smooth. A small horizontal force is now applied to the trolley in the forward direction of motion and gradually increased.
3. Determine whether the block will topple or slide.

6 A sack of beans of mass 40 kg is pulled from rest at point A up a non-uniform slope onto and along a horizontal platform. Fig. 6 shows this slope AB and the platform BC , which is a vertical distance of 12 m above A. \begin{figure}[h] \end{figure}

1. Calculate the gain in the gravitational potential energy of the sack when it is moved from A to the platform. The sack has a speed of \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) by the time it reaches C at the far end of the platform. The total work done against friction in moving the sack from A to C is 484 J . There are no other resistances to the sack's motion.
2. Calculate the total work done in moving the sack between the points A and C . At point C , travelling at \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the sack starts to slide down a straight chute inclined at \(\alpha\) to the horizontal. Point D at the bottom of the chute is at the same vertical height as A , as shown in Fig. 6. The chute is rough and the coefficient of friction between the chute and the sack is 0.6 . During this part of the motion, again the only resistance to the motion of the sack is friction.
3. Use an energy method to calculate the value of \(\alpha\) given that the sack is travelling at \(3 \mathrm {~ms} ^ { - 1 }\) when it reaches D . For safety reasons the sack needs to arrive at D with a speed of less than \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The value of \(\alpha\) can be adjusted to try to achieve this.
4. (A) Find the range of values of \(\alpha\) which achieve a safe speed at D .
   (B) Comment on whether adjusting \(\alpha\) is a practical way of achieving a safe speed at D .

9. The diagram below shows a parcel, of mass \(m \mathrm {~kg}\), sliding down a rough slope inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 7 } { 25 }\). \includegraphics[max width=\textwidth, alt={}, center]{8f47b2ff-f954-42ec-8ecc-fc64313a7b89-24_394_906_497_584} The coefficient of friction between the parcel and the slope is \(\frac { 1 } { 12 }\). In addition to friction, the parcel experiences a variable resistive force of \(m v \mathrm {~N}\), where \(v \mathrm {~ms} ^ { - 1 }\) is the velocity of the parcel at time \(t\) seconds.

1. Show that the motion of the parcel satisfies the differential equation $$5 \frac { \mathrm {~d} v } { \mathrm {~d} t } = g - 5 v$$

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6. A slope is inclined at an angle of \(5 ^ { \circ }\) to the horizontal. A car, of mass 1500 kg , has an engine that is working at a constant rate of \(P \mathrm {~W}\). The resistance to motion of the car is constant at 4500 N . When the car is moving up the slope, its acceleration is \(a \mathrm {~ms} ^ { - 2 }\) at the instant when its speed is \(10 \mathrm {~ms} ^ { - 1 }\). When the car is moving down the slope, its deceleration is \(a \mathrm {~ms} ^ { - 2 }\) at the instant when its speed is \(20 \mathrm {~ms} ^ { - 1 }\). Determine the value of \(P\) and the value of \(a\). \includegraphics[max width=\textwidth, alt={}, center]{ae23a093-1419-4be4-8285-951650dc5a35-14_87_1609_635_267}

7 In this question use \(g = 9.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
When a car, of mass 1200 kg , travels at a speed of \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) it experiences a total resistive force which can be modelled as being of magnitude \(36 v\) newtons.
The maximum power of the car is 90 kilowatts.
The car starts to descend a hill, inclined at \(5.2 ^ { \circ }\) to the horizontal, along a straight road.
Find the maximum speed of the car down this hill.
[0pt] [5 marks]

1. \begin{figure}[h] \end{figure} A small book of mass \(m\) is held on a rough straight desk lid which is inclined at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac { 3 } { 4 }\). The book is released from rest at a distance of 0.5 m from the edge of the desk lid, as shown in Figure 1. The book slides down the desk lid and then hits the floor that is 0.8 m below the edge of the desk lid. The coefficient of friction between the book and the desk lid is 0.4 The book is modelled as a particle which, after leaving the desk lid, is assumed to move freely under gravity.

1. Find, in terms of \(m\) and \(g\), the magnitude of the normal reaction on the book as it slides down the desk lid.
2. Use the work-energy principle to find the speed of the book as it hits the floor.

1. A plane is inclined to the horizontal at an angle \(\alpha\), where \(\tan \alpha = \frac { 3 } { 4 }\)

A particle \(P\) is held at rest at a point \(A\) on the plane.
The particle \(P\) is then projected with speed \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from \(A\), up a line of greatest slope of the plane. In an initial model, the plane is modelled as being smooth and air resistance is modelled as being negligible. Using this model and the principle of conservation of mechanical energy,

1. find the speed of \(P\) at the instant when it has travelled a distance \(\frac { 25 } { 6 } \mathrm {~m}\) up the plane from \(A\). In a refined model, the plane is now modelled as being rough, with the coefficient of friction between \(P\) and the plane being \(\frac { 3 } { 5 }\) Air resistance is still modelled as being negligible.
   Using this refined model and the work-energy principle,
2. find the speed of \(P\) at the instant when it has travelled a distance \(\frac { 25 } { 6 } \mathrm {~m}\) up the plane from \(A\).

3. \begin{figure}[h] \end{figure} Figure 1 shows part of the end elevation of a building which sits on horizontal ground. The side of the building is vertical and has height \(h\). A small stone of mass \(m\) is at rest on the roof of the building at the point \(A\). The stone slides from rest down a line of greatest slope of the roof and reaches the edge \(B\) of the roof with speed \(\sqrt { 2 g h }\) The stone then moves under gravity before hitting the ground with speed \(W\).
In a model of the motion of the stone from \(\boldsymbol { B }\) to the ground

• the stone is modelled as a particle
• air resistance is ignored

Using the principle of conservation of mechanical energy and the model,

1. find \(W\) in terms of \(g\) and \(h\). In a model of the motion of the stone from \(\boldsymbol { A }\) to \(\boldsymbol { B }\)
   Using this model,
2. find, in terms of \(m\) and \(g\), the magnitude of the frictional force acting on the stone as it slides down the roof,
3. use the work-energy principle to find \(d\) in terms of \(h\).

1. A light elastic string with natural length \(l\) and modulus of elasticity \(k m g\) has one end attached to a fixed point \(A\) on a rough inclined plane. The other end of the string is attached to a package of mass \(m\).

The plane is inclined at an angle \(\theta\) to the horizontal, where \(\tan \theta = \frac { 5 } { 12 }\) The package is initially held at \(A\). The package is then projected with speed \(\sqrt { 6 g l }\) up a line of greatest slope of the plane and first comes to rest at the point \(B\), where \(A B = 31\).
The coefficient of friction between the package and the plane is \(\frac { 1 } { 4 }\) By modelling the package as a particle,

1. show that \(k = \frac { 15 } { 26 }\)
2. find the acceleration of the package at the instant it starts to move back down the plane from the point \(B\).

6. \begin{figure}[h] \end{figure} A light elastic spring has natural length \(3 l\) and modulus of elasticity \(3 m g\).
One end of the spring is attached to a fixed point \(X\) on a rough inclined plane.
The other end of the spring is attached to a package \(P\) of mass \(m\).
The plane is inclined to the horizontal at an angle \(\alpha\) where \(\tan \alpha = \frac { 3 } { 4 }\) The package is initially held at the point \(Y\) on the plane, where \(X Y = l\). The point \(Y\) is higher than \(X\) and \(X Y\) is a line of greatest slope of the plane, as shown in Figure 2. The package is released from rest at \(Y\) and moves up the plane.
The coefficient of friction between \(P\) and the plane is \(\frac { 1 } { 3 }\) By modelling \(P\) as a particle,

1. show that the acceleration of \(P\) at the instant when \(P\) is released from rest is \(\frac { 17 } { 15 } \mathrm {~g}\)
2. find, in terms of \(g\) and \(l\), the speed of \(P\) at the instant when the spring first reaches its natural length of 31 .

6. \begin{figure}[h] \end{figure} Two blocks, \(A\) and \(B\), of masses 2 kg and 4 kg respectively are attached to the ends of a light inextensible string. Initially \(A\) is held on a fixed rough plane. The plane is inclined to horizontal ground at an angle \(\theta\), where \(\tan \theta = \frac { 3 } { 4 }\) The string passes over a small smooth light pulley \(P\) that is fixed at the top of the plane. The part of the string from \(A\) to \(P\) is parallel to a line of greatest slope of the plane. Block \(A\) is held on the plane with the distance \(A P\) greater than 3 m .
Block \(B\) hangs freely below \(P\) at a distance of 3 m above the ground, as shown in Figure 4. The coefficient of friction between \(A\) and the plane is \(\mu\) Block \(A\) is released from rest with the string taut.
By modelling the blocks as particles,

1. find the potential energy lost by the whole system as a result of \(B\) falling 3 m . Given that the speed of \(B\) at the instant it hits the ground is \(4.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and ignoring air resistance,
2. use the work-energy principle to find the value of \(\mu\) After \(B\) hits the ground, \(A\) continues to move up the plane but does not reach the pulley in the subsequent motion.
   Block \(A\) comes to instantaneous rest after moving a total distance of ( \(3 + d\) ) m from its point of release. Ignoring air resistance,
3. use the work-energy principle to find the value of \(d\) \includegraphics[max width=\textwidth, alt={}, center]{86a37170-046f-46e5-9c8c-06d5f98ca4fe-20_2255_50_309_1981}

1. A rough plane is inclined to the horizontal at an angle \(\theta\), where \(\tan \theta = \frac { 3 } { 4 }\)

A particle \(P\) of mass \(m\) is at rest at a point on the plane. The particle is projected up the plane with speed \(\sqrt { 2 a g }\) The particle moves up a line of greatest slope of the plane and comes to instantaneous rest after moving a distance \(d\). The coefficient of friction between \(P\) and the plane is \(\frac { 1 } { 7 }\)

1. Show that the magnitude of the frictional force acting on \(P\) as it moves up the plane is \(\frac { 4 m g } { 35 }\) Air resistance is assumed to be negligible.
   Using the work-energy principle,
2. find \(d\) in terms of \(a\).

1. A parcel of mass 5 kg is projected with speed \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) up a line of greatest slope of a fixed rough inclined ramp.
   The ramp is inclined at angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 1 } { 7 }\) The parcel is projected from the point \(A\) on the ramp and comes to instantaneous rest at the point \(B\) on the ramp, where \(A B = 14 \mathrm {~m}\).

The coefficient of friction between the parcel and the ramp is \(\mu\).
In a model of the parcel's motion, the parcel is treated as a particle.

1. Use the work-energy principle to find the value of \(\mu\).
2. Suggest one way in which the model could be refined to make it more realistic.

1. Use an energy method to find the magnitude of the frictional force acting on the block. Calculate the coefficient of friction between the block and the plane.
2. Calculate the power of the tension in the string when the block has a speed of \(7 \mathrm {~ms} ^ { - 1 }\). Fig. 3.1 shows a thin planar uniform rigid rectangular sheet of metal, OPQR, of width 1.65 m and height 1.2 m . The mass of the sheet is \(M \mathrm {~kg}\). The sides OP and PQ have thin rigid uniform reinforcements attached with masses \(0.6 M \mathrm {~kg}\) and \(0.4 M \mathrm {~kg}\), respectively. Fig. 3.1 also shows coordinate axes with origin at O . The sheet with its reinforcements is to be used as an inn sign.

1. Calculate the coordinates of the centre of mass of the inn sign. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{8fb49c8b-92e5-49e5-9a3a-e8391c82d9a1-4_421_492_210_1334} \captionsetup{labelformat=empty} \caption{Fig. 3.1}
   \end{figure} The inn sign has a weight of 300 N . It hangs in equilibrium with QR horizontal when vertical forces \(Y _ { \mathrm { Q } } \mathrm { N }\) and \(Y _ { \mathrm { R } } \mathrm { N }\) act at Q and R respectively.
2. Calculate the value of \(Y _ { \mathrm { Q } }\) and show that \(Y _ { \mathrm { R } } = 120\). The inn sign is hung from a framework, ABCD , by means of two light vertical inextensible wires attached to the sign at Q and R and the framework at B and C , as shown in Fig. 3.2. QR and BC are horizontal. The framework is made from light rigid rods \(\mathrm { AB } , \mathrm { BC } , \mathrm { CA }\) and CD freely pin-jointed together at \(\mathrm { A } , \mathrm { B }\) and C and to a vertical wall at A and D . Fig. 3.3 shows the dimensions of the framework in metres as well as the external forces \(X _ { \mathrm { A } } \mathrm { N } , Y _ { \mathrm { A } } \mathrm { N }\) acting at A and \(X _ { \mathrm { D } } \mathrm { N } , Y _ { \mathrm { D } } \mathrm { N }\) acting at D . You are given that \(\sin \alpha = \frac { 5 } { 13 } , \cos \alpha = \frac { 12 } { 13 } , \sin \beta = \frac { 4 } { 5 }\) and \(\cos \beta = \frac { 3 } { 5 }\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{8fb49c8b-92e5-49e5-9a3a-e8391c82d9a1-4_543_526_1420_253} \captionsetup{labelformat=empty} \caption{Fig. 3.2}
   \end{figure} \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{8fb49c8b-92e5-49e5-9a3a-e8391c82d9a1-4_629_793_1343_964} \captionsetup{labelformat=empty} \caption{Fig. 3.3}
   \end{figure}
3. Mark on the diagram in your Printed Answer Book all the forces acting on the pin-joints at \(\mathrm { A } , \mathrm { B } , \mathrm { C }\) and D , including those internal to the rods, when the inn sign is hanging from the framework.
4. Show that \(X _ { \mathrm { D } } = 261\).
5. Calculate the forces internal to the rods \(\mathrm { AB } , \mathrm { BC }\) and CD , stating whether each rod is in tension or thrust (compression). Calculate also the values of \(Y _ { \mathrm { D } }\) and \(Y _ { \mathrm { A } }\). [Your working in this part should correspond to your diagram in part (iii).]

1 A train consists of a locomotive pulling 17 identical trucks. The mass of the locomotive is 120 tonnes and the mass of each truck is 40 tonnes. The locomotive gives a driving force of 121000 N . The resistance to motion on each truck is \(R \mathrm {~N}\) and the resistance on the locomotive is \(5 R \mathrm {~N}\).
Initially the train is travelling on a straight horizontal track and its acceleration is \(0.11 \mathrm {~ms} ^ { - 2 }\).

1. Show that \(R = 1500\).
2. Find the tensions in the couplings between
   (A) the last two trucks,
   (B) the locomotive and the first truck. The train now comes to a place where the track goes up a straight, uniform slope at an angle \(\alpha\) with the horizontal, where \(\sin \alpha = \frac { 1 } { 80 }\). The driving force and the resistance forces remain the same as before.
3. Find the magnitude and direction of the acceleration of the train. The train then comes to a straight uniform downward slope at an angle \(\beta\) to the horizontal.
   The driver of the train reduces the driving force to zero and the resistance forces remain the same as before. The train then travels at a constant speed down the slope.
4. Find the value of \(\beta\).