6.02i Conservation of energy: mechanical energy principle

4 \includegraphics[max width=\textwidth, alt={}, center]{155bc571-80e4-4c93-859f-bb150a109211-3_489_1041_258_552} \(A B C\) is a vertical cross-section of a surface. The part of the surface containing \(A B\) is smooth and \(A\) is 4 m higher than \(B\). The part of the surface containing \(B C\) is horizontal and the distance \(B C\) is 5 m (see diagram). A particle of mass 0.8 kg is released from rest at \(A\) and slides along \(A B C\). Find the speed of the particle at \(C\) in each of the following cases.

1. The horizontal part of the surface is smooth.
2. The coefficient of friction between the particle and the horizontal part of the surface is 0.3 .

6 \includegraphics[max width=\textwidth, alt={}, center]{2bb3c9bb-60f0-440d-a148-b4db3478ca31-3_382_1451_797_347} The diagram shows the vertical cross-section \(A B C D\) of a surface. \(B C\) is a circular arc, and \(A B\) and \(C D\) are tangents to \(B C\) at \(B\) and \(C\) respectively. \(A\) and \(D\) are at the same horizontal level, and \(B\) and \(C\) are at heights 2.7 m and 3.0 m respectively above the level of \(A\) and \(D\). A particle \(P\) of mass 0.2 kg is given a velocity of \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at \(A\), in the direction of \(A B\) (see diagram). The parts of the surface containing \(A B\) and \(B C\) are smooth.

1. Find the decrease in the speed of \(P\) as \(P\) moves along the surface from \(B\) to \(C\). The part of the surface containing \(C D\) exerts a constant frictional force on \(P\), as it moves from \(C\) to \(D\), and \(P\) comes to rest as it reaches \(D\).
2. Find the speed of \(P\) when it is at the mid-point of \(C D\).

2 \includegraphics[max width=\textwidth, alt={}, center]{fd534430-2619-4078-ad0a-2355e656e121-2_569_519_676_813} Particle \(A\) of mass 0.2 kg and particle \(B\) of mass 0.6 kg are attached to the ends of a light inextensible string. The string passes over a fixed smooth pulley. \(B\) is held at rest at a height of 1.6 m above the floor. \(A\) hangs freely at a height of \(h \mathrm {~m}\) above the floor. Both straight parts of the string are vertical (see diagram). \(B\) is released and both particles start to move. When \(B\) reaches the floor it remains at rest, but \(A\) continues to move vertically upwards until it reaches a height of 3 m above the floor. Find the speed of \(B\) immediately before it hits the floor, and hence find the value of \(h\).

7 \includegraphics[max width=\textwidth, alt={}, center]{ffefbc81-402f-4048-8741-23c8bae30d5a-3_246_1006_1781_571} A block of mass 60 kg is pulled up a hill in the line of greatest slope by a force of magnitude 50 N acting at an angle \(\alpha ^ { \circ }\) above the hill. The block passes through points \(A\) and \(B\) with speeds \(8.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(3.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively (see diagram). The distance \(A B\) is 250 m and \(B\) is 17.5 m above the level of \(A\). The resistance to motion of the block is 6 N . Find the value of \(\alpha\).
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2 A particle of mass 0.5 kg starts from rest and slides down a line of greatest slope of a smooth plane. The plane is inclined at an angle of \(30 ^ { \circ }\) to the horizontal.

1. Find the time taken for the particle to reach a speed of \(2.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). When the particle has travelled 3 m down the slope from its starting point, it reaches rough horizontal ground at the bottom of the slope. The frictional force acting on the particle is 1 N .
2. Find the distance that the particle travels along the ground before it comes to rest.

4 \includegraphics[max width=\textwidth, alt={}, center]{48f66bd5-33c1-4ce9-85f9-69faf10e871c-3_574_483_260_829} The diagram shows a vertical cross-section \(A B C\) of a surface. The part of the surface containing \(A B\) is smooth and \(A\) is 2.5 m above the level of \(B\). The part of the surface containing \(B C\) is rough and is at \(45 ^ { \circ }\) to the horizontal. The distance \(B C\) is 4 m (see diagram). A particle \(P\) of mass 0.2 kg is released from rest at \(A\) and moves in contact with the curve \(A B\) and then with the straight line \(B C\). The coefficient of friction between \(P\) and the part of the surface containing \(B C\) is 0.4 . Find the speed with which \(P\) reaches \(C\).

7 A straight hill \(A B\) has length 400 m with \(A\) at the top and \(B\) at the bottom and is inclined at an angle of \(4 ^ { \circ }\) to the horizontal. A straight horizontal road \(B C\) has length 750 m . A car of mass 1250 kg has a speed of \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at \(A\) when starting to move down the hill. While moving down the hill the resistance to the motion of the car is 2000 N and the driving force is constant. The speed of the car on reaching \(B\) is \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. By using work and energy, find the driving force of the car. On reaching \(B\) the car moves along the road \(B C\). The driving force is constant and twice that when the car was on the hill. The resistance to the motion of the car continues to be 2000 N . Find
2. the acceleration of the car while moving from \(B\) to \(C\),
3. the power of the car's engine as the car reaches \(C\).

5 \includegraphics[max width=\textwidth, alt={}, center]{a92f97e2-343f-4cac-ae38-f18a4ad49055-3_574_1205_260_470} The diagram shows a velocity-time graph which models the motion of a cyclist. The graph consists of five straight line segments. The cyclist accelerates from rest to a speed of \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) over a period of 10 s , and then travels at this speed for a further 20 s . The cyclist then descends a hill, accelerating to speed \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) over a period of 10 s . This speed is maintained for a further 30 s . The cyclist then decelerates to rest over a period of 20 s .

1. Find the acceleration of the cyclist during the first 10 seconds.
2. Show that the total distance travelled by the cyclist in the 90 seconds of motion may be expressed as \(( 45 V + 150 ) \mathrm { m }\). Hence find \(V\), given that the total distance travelled by the cyclist is 465 m .
3. The combined mass of the cyclist and the bicycle is 80 kg . The cyclist experiences a constant resistance to motion of 20 N . Use an energy method to find the vertical distance which the cyclist descends during the downhill section from \(t = 30\) to \(t = 40\), assuming that the cyclist does no work during this time.

6 A block of mass 25 kg is pulled along horizontal ground by a force of magnitude 50 N inclined at \(10 ^ { \circ }\) above the horizontal. The block starts from rest and travels a distance of 20 m . There is a constant resistance force of magnitude 30 N opposing motion.

1. Find the work done by the pulling force.
2. Use an energy method to find the speed of the block when it has moved a distance of 20 m .
3. Find the greatest power exerted by the 50 N force. \includegraphics[max width=\textwidth, alt={}, center]{a92f97e2-343f-4cac-ae38-f18a4ad49055-3_236_1027_2161_566} After the block has travelled the 20 m , it comes to a plane inclined at \(5 ^ { \circ }\) to the horizontal. The force of 50 N is now inclined at an angle of \(10 ^ { \circ }\) to the plane and pulls the block directly up the plane (see diagram). The resistance force remains 30 N .
4. Find the time it takes for the block to come to rest from the instant when it reaches the foot of the inclined plane.
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4 A girl on a sledge starts, with a speed of \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), at the top of a slope of length 100 m which is at an angle of \(20 ^ { \circ }\) to the horizontal. The sledge slides directly down the slope.

1. Given that there is no resistance to the sledge's motion, find the speed of the sledge at the bottom of the slope.
2. It is given instead that the sledge experiences a resistance to motion such that the total work done against the resistance is 8500 J , and the speed of the sledge at the bottom of the slope is \(21 \mathrm {~ms} ^ { - 1 }\). Find the total mass of the girl and the sledge.

7 A box of mass 50 kg is at rest on a plane inclined at \(10 ^ { \circ }\) to the horizontal.

1. Find an inequality for the coefficient of friction between the box and the plane. In fact the coefficient of friction between the box and the plane is 0.19 .
2. A girl pushes the box with a force of 50 N , acting down a line of greatest slope of the plane, for a distance of 5 m . She then stops pushing. Use an energy method to find the speed of the box when it has travelled a further 5 m . The box then comes to a plane inclined at \(20 ^ { \circ }\) below the horizontal. The box moves down a line of greatest slope of this plane. The coefficient of friction is still 0.19 and the girl is not pushing the box.
3. Find the acceleration of the box.

3 A roller-coaster car (including passengers) has a mass of 840 kg . The roller-coaster ride includes a section where the car climbs a straight ramp of length 8 m inclined at \(30 ^ { \circ }\) above the horizontal. The car then immediately descends another ramp of length 10 m inclined at \(20 ^ { \circ }\) below the horizontal. The resistance to motion acting on the car is 640 N throughout the motion.

1. Find the total work done against the resistance force as the car ascends the first ramp and descends the second ramp.
2. The speed of the car at the bottom of the first ramp is \(14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Use an energy method to find the speed of the car when it reaches the bottom of the second ramp.

3 A van of mass 2500 kg descends a hill of length 0.4 km inclined at \(4 ^ { \circ }\) to the horizontal. There is a constant resistance to motion of 600 N and the speed of the van increases from \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) as it descends the hill. Find the work done by the van's engine as it descends the hill.

4 Two particles \(A\) and \(B\), of masses \(m \mathrm {~kg}\) and 0.3 kg respectively, are attached to the ends of a light inextensible string. The string passes over a fixed smooth pulley and the particles hang freely below it. The system is released from rest, with both particles 0.8 m above horizontal ground. Particle \(A\) reaches the ground with a speed of \(0.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Find the tension in the string during the motion before \(A\) reaches the ground.
2. Find the value of \(m\).

3 A particle of mass 1.2 kg moves in a straight line \(A B\). It is projected with speed \(7.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from \(A\) towards \(B\) and experiences a resistance force. The work done against this resistance force in moving from \(A\) to \(B\) is 25 J .

1. Given that \(A B\) is horizontal, find the speed of the particle at \(B\).
2. It is given instead that \(A B\) is inclined at \(30 ^ { \circ }\) below the horizontal and that the speed of the particle at \(B\) is \(9 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The work done against the resistance force remains the same. Find the distance \(A B\).

6 A van of mass 3200 kg travels along a horizontal road. The power of the van's engine is constant and equal to 36 kW , and there is a constant resistance to motion acting on the van.

1. When the speed of the van is \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), its acceleration is \(0.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Find the resistance force.
   When the van is travelling at \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), it begins to ascend a hill inclined at \(1.5 ^ { \circ }\) to the horizontal. The power is increased and the resistance force is still equal to the value found in part (i).
2. Find the power required to maintain this speed of \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
3. The engine is now stopped, with the van still travelling at \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), and the van decelerates to rest. Find the distance the van moves up the hill from the point at which the engine is stopped until it comes to rest.

1 One end of a light elastic string of natural length 1.6 m and modulus of elasticity 25 N is attached to a fixed point \(A\). A particle \(P\) of mass 0.15 kg is attached to the other end of the string. \(P\) is held at rest at a point 2 m vertically below \(A\) and is then released.

1. For the motion from the instant of release until the string becomes slack, find the loss of elastic potential energy and the gain in gravitational potential energy.
2. Hence find the speed of \(P\) at the instant the string becomes slack.

7 \includegraphics[max width=\textwidth, alt={}, center]{7f8646df-a7d8-4ca1-a6ee-3ceab6bb83af-4_232_905_762_621} A light elastic string has natural length 10 m and modulus of elasticity 130 N . The ends of the string are attached to fixed points \(A\) and \(B\), which are at the same horizontal level. A small stone is attached to the mid-point of the string and hangs in equilibrium at a point 2.5 m below \(A B\), as shown in the diagram. With the stone in this position the length of the string is 13 m .

1. Find the tension in the string.
2. Show that the mass of the stone is 3 kg . The stone is now held at rest at a point 8 m vertically below the mid-point of \(A B\).
3. Find the elastic potential energy of the string in this position.
4. The stone is now released. Find the speed with which it passes through the mid-point of \(A B\).

5 \includegraphics[max width=\textwidth, alt={}, center]{835616aa-0b2b-4e8c-bbbf-60b72dc5ea3e-3_321_698_1692_726} One end of a light elastic string of natural length 4 m and modulus of elasticity 200 N is attached to a fixed point \(A\). The other end is attached to the end \(C\) of a uniform rod \(C D\) of mass 10 kg . One end of another light elastic string, which is identical to the first, is attached to a fixed point \(B\) and the other end is attached to \(D\), as shown in the diagram. The distance \(A B\) is equal to the length of the rod, and \(A B\) is horizontal. The rod is released from rest with \(C\) at \(A\) and \(D\) at \(B\). While the strings are taut, the speed of the rod is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when the rod is at a distance of \(( 4 + x ) \mathrm { m }\) below \(A B\).

1. Show that \(v ^ { 2 } = 10 \left( 8 + 2 x - x ^ { 2 } \right)\).
2. Hence find the value of \(x\) when the rod is at its lowest point.

4 A particle \(P\) of mass 0.4 kg is attached to one end of a light elastic string of natural length 1.5 m and modulus of elasticity 6 N . The other end of the string is attached to a fixed point \(O\) on a rough horizontal table. \(P\) is released from rest at a point on the table 3.5 m from \(O\). The speed of \(P\) at the instant the string becomes slack is \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find

1. the work done against friction during the period from the release of \(P\) until the string becomes slack,
2. the coefficient of friction between \(P\) and the table.

6 A light elastic string has natural length 2 m and modulus of elasticity 0.8 N . One end of the string is attached to a fixed point \(O\) of a rough plane which is inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 12 } { 13 }\). A particle \(P\) of mass 0.052 kg is attached to the other end of the string. The coefficient of friction between the particle and the plane is 0.4 . \(P\) is released from rest at \(O\).

1. When \(P\) has moved \(d\) metres down the plane from \(O\), where \(d > 2\), find expressions in terms of \(d\) for
   1. the loss in gravitational potential energy of \(P\),
   2. the gain in elastic potential energy of the string,
   3. the work done by the frictional force acting on \(P\).
   4. Show that \(d ^ { 2 } - 6 d + 4 = 0\) when \(P\) is at its lowest point, and hence find the value of \(d\) in this case.

5 One end of a light elastic string, of natural length 0.5 m and modulus of elasticity 140 N , is attached to a fixed point \(O\). A particle of mass 0.8 kg is attached to the other end of the string. The particle is released from rest at \(O\). By considering the energy of the system, find

1. the speed of the particle when the extension of the string is 0.1 m ,
2. the extension of the string when the particle is at its lowest point.

6 \includegraphics[max width=\textwidth, alt={}, center]{57f7ca89-f028-447a-9ac9-55f931201e6b-3_83_771_1978_689} \(A\) and \(B\) are fixed points on a smooth horizontal table. The distance \(A B\) is 2.5 m . An elastic string of natural length 0.6 m and modulus of elasticity 24 N has one end attached to the table at \(A\), and the other end attached to a particle \(P\) of mass 0.95 kg . Another elastic string of natural length 0.9 m and modulus of elasticity 18 N has one end attached to the table at \(B\), and the other end attached to \(P\). The particle \(P\) is held at rest at the mid-point of \(A B\) (see diagram).

1. Find the tensions in the strings. The particle is released from rest.
2. Find the acceleration of \(P\) immediately after its release.
3. \(P\) reaches its maximum speed at the point \(C\). Find the distance \(A C\).

6 One end of a light elastic string of natural length 1.25 m and modulus of elasticity 20 N is attached to a fixed point \(O\). A particle \(P\) of mass 0.5 kg is attached to the other end of the string. \(P\) is held at rest at \(O\) and then released. When the extension of the string is \(x \mathrm {~m}\) the speed of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Show that \(v ^ { 2 } = - 32 x ^ { 2 } + 20 x + 25\).
2. Find the maximum speed of \(P\).
3. Find the acceleration of \(P\) when it is at its lowest point.

6 \includegraphics[max width=\textwidth, alt={}, center]{fb79f949-567c-4dbb-8533-7b7278cad21c-3_200_639_1754_753} A particle \(P\) of mass 1.6 kg is attached to one end of each of two light elastic strings. The other ends of the strings are attached to fixed points \(A\) and \(B\) which are 2 m apart on a smooth horizontal table. The string attached to \(A\) has natural length 0.25 m and modulus of elasticity 4 N , and the string attached to \(B\) has natural length 0.25 m and modulus of elasticity 8 N . The particle is held at the mid-point \(M\) of \(A B\) (see diagram).

1. Find the tensions in the strings.
2. Show that the total elastic potential energy in the two strings is 13.5 J . \(P\) is released from rest and in the subsequent motion both strings remain taut. The displacement of \(P\) from \(M\) is denoted by \(x \mathrm {~m}\). Find
3. the initial acceleration of \(P\),
4. the non-zero value of \(x\) at which the speed of \(P\) is zero. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{fb79f949-567c-4dbb-8533-7b7278cad21c-4_529_542_269_804} \captionsetup{labelformat=empty} \caption{Fig. 1}
   \end{figure} A uniform solid body has a cross-section as shown in Fig. 1.
5. Show that the centre of mass of the body is 2.5 cm from the plane face containing \(O B\) and 3.5 cm from the plane face containing \(O A\).
6. The solid is placed on a rough plane which is initially horizontal. The coefficient of friction between the solid and the plane is \(\mu\).
   1. \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{fb79f949-567c-4dbb-8533-7b7278cad21c-4_332_469_1320_918} \captionsetup{labelformat=empty} \caption{Fig. 2}
      \end{figure} The solid is placed with \(O A\) in contact with the plane, and then the plane is tilted so that \(O A\) lies along a line of greatest slope with \(A\) higher than \(O\) (see Fig. 2). When the angle of inclination is sufficiently great the solid starts to topple (without sliding). Show that \(\mu > \frac { 5 } { 7 }\).
      [0pt] [5]
   2. \includegraphics[max width=\textwidth, alt={}, center]{fb79f949-567c-4dbb-8533-7b7278cad21c-4_291_465_1987_918} Instead, the solid is placed with \(O B\) in contact with the plane, and then the plane is tilted so that \(O B\) lies along a line of greatest slope with \(B\) higher than \(O\) (see Fig. 3). When the angle of inclination is sufficiently great the solid starts to slide (without toppling). Find another inequality for \(\mu\).