6.02i Conservation of energy: mechanical energy principle

8 In this question use \(g = 10 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
A particle, of mass 2 kg , is attached to one end of a light elastic string of natural length 0.2 metres. The other end of the string is attached to a fixed point \(O\).
The particle is pulled down and released from rest at a point 0.6 metres directly below \(O\).
The particle then moves vertically and next comes to rest when it is 0.1 metres below \(O\).
Assume that no air resistance acts on the particle.
8

1. Find the maximum speed of the particle.
   [0pt] [6 marks]
   8
2. Describe one way in which the model you have used could be refined.

1
A particle \(P\) of mass \(m \mathrm {~kg}\) is attached to one end of a light inextensible string of length 3.2 m . The other end of the string is attached to a fixed point \(O\). The particle is held at rest, with the string taut and making an angle of \(15 ^ { \circ }\) with the vertical. It is then projected with velocity \(1.2 \mathrm {~ms} ^ { - 1 }\) in a direction perpendicular to \(O P\) and with a downwards component so that it begins to move in a vertical circle (see diagram). In the ensuing motion the string remains taut and the angle it makes with the downwards vertical through \(O\) is denoted by \(\theta ^ { \circ }\).

1. Find the speed of \(P\) at the point on its path vertically below \(O\).
2. Find the value of \(\theta\) at the point where \(P\) first comes to instantaneous rest.

1 \includegraphics[max width=\textwidth, alt={}, center]{74bada9e-60cf-4ed4-abd0-4be155b7cf81-2_533_424_402_246} A smooth wire is shaped into a circle of radius 2.5 m which is fixed in a vertical plane with its centre at a point \(O\). A small bead \(B\) is threaded onto the wire. \(B\) is held with \(O B\) vertical and is then projected horizontally with an initial speed of \(8.4 \mathrm {~ms} ^ { - 1 }\) (see diagram).

1. Find the speed of \(B\) at the instant when \(O B\) makes an angle of 0.8 radians with the downward vertical through \(O\).
2. Determine whether \(B\) has sufficient energy to reach the point on the wire vertically above \(O\).

2 A hockey puck of mass 0.2 kg is sliding down a rough slope which is inclined at \(10 ^ { \circ }\) to the horizontal. At the instant that its velocity is \(14 \mathrm {~ms} ^ { - 1 }\) directly down the slope it is hit by a hockey stick. Immediately after it is hit its velocity is \(24 \mathrm {~ms} ^ { - 1 }\) directly up the slope.

1. Find the magnitude of the impulse exerted by the hockey stick on the puck. After it has been hit, the puck first comes to instantaneous rest when it has travelled 15 m up the slope. While the puck is moving up the slope, the resistance to its motion has constant magnitude \(R \mathrm {~N}\).
2. Use an energy method to determine the value of \(R\).

3 A smooth wire is shaped into a circle of radius 4.2 m which is fixed in a vertical plane with its centre at a point \(O\). A small bead \(B\) is threaded onto the wire. \(B\) is held so that \(O B\) makes an angle of \(\frac { 1 } { 3 } \pi\) radians with the downwards vertical through \(O\). \(B\) is projected downwards along the wire with initial speed \(u \mathrm {~ms} ^ { - 1 }\) (see diagram). In its subsequent motion \(B\) describes complete circles about \(O\). \includegraphics[max width=\textwidth, alt={}, center]{98053e88-1aec-4b0d-ae5f-ece4ad340266-3_493_665_561_242} Given that the lowest speed of \(B\) in its motion is \(4 \mathrm {~ms} ^ { - 1 }\) determine the value of \(u\).

8 As part of an industrial process a single pump causes the intake of a liquid chemical to the bottom end of a tube, draws it up the tube and then discharges it through a nozzle at the top end of the tube. The tube is straight and narrow, 35 m long and inclined at an angle of \(26 ^ { \circ }\) to the horizontal. The chemical arrives at the intake at the bottom end of the tube with a speed of \(6.2 \mathrm {~ms} ^ { - 1 }\). At the top end of the tube the chemical is discharged horizontally with a speed of \(14.3 \mathrm {~ms} ^ { - 1 }\) (see diagram). In total, the pump discharges 1500 kg of chemical through the nozzle each hour. \includegraphics[max width=\textwidth, alt={}, center]{98053e88-1aec-4b0d-ae5f-ece4ad340266-5_405_1175_685_242} In order to model the changes to the mechanical energy of the chemical during the entire process of intake, drawing and discharge, the following modelling assumptions are made.

• At any instant the total resistance to the motion of all the liquid in the tube is 40 N .
• All other resistances to motion are ignored.
• The liquid in the tube moves at a constant speed of \(6.2 \mathrm {~ms} ^ { - 1 }\).
  1. 1. Find the difference between the total amount of energy output by the pump each hour and the total amount of mechanical energy gained by the chemical each hour.
     2. Give one reason why the model underestimates the power of the engine.

2 \includegraphics[max width=\textwidth, alt={}, center]{b190b8c9-75b0-4ede-913f-cdecdb58180f-2_337_579_842_246} A small body \(P\) of mass 3 kg is at rest at the lowest point of the inside of a smooth hemispherical shell of radius 3.2 m and centre \(O\). \(P\) is projected horizontally with a speed of \(u \mathrm {~ms} ^ { - 1 }\). When \(P\) first comes to instantaneous rest \(O P\) makes an angle of \(60 ^ { \circ }\) with the downward vertical through \(O\).

1. Find the value of \(u\).
2. State one assumption made in modelling the motion of \(P\).

2 A particle \(P\) of mass 0.4 kg is attached to one end of a light inextensible string of length 1.8 m . The other end of the string is attached to a fixed point, \(O\), on a smooth horizontal plane. Initially, \(P\) is moving with a constant speed of \(12 \mathrm {~ms} ^ { - 1 }\) in a horizontal circle with \(O\) as its centre.

1. 1. Find the magnitude of the acceleration of \(P\).
   2. State the direction of the acceleration of \(P\). A force is now applied to \(P\) in such a way that its angular velocity increases. At the instant that the angular velocity reaches \(8 \mathrm { rad } \mathrm { s } ^ { - 1 }\), the string breaks.
2. 1. Find the speed with which \(P\) is moving at the instant that the string breaks.
   2. Find the tension in the string at the instant that the string breaks. After the string has broken \(P\) starts to move directly up a smooth slope which is fixed to the plane and inclined at an angle \(\theta ^ { \circ }\) above the horizontal. Particle \(P\) moves a distance of 20 m up the slope before coming to instantaneous rest.
3. Use an energy method to determine the value of \(\theta\).

7 A particle \(P\) of mass 3.5 kg is attached to one end of a rod of length 5.4 m . The other end of the rod is hinged at a fixed point \(O\) and \(P\) hangs in equilibrium directly below \(O\). A horizontal impulse of magnitude 44.1 Ns is applied to \(P\).
In an initial model of the subsequent motion of \(P\) the rod is modelled as being light and inextensible and all resistance to the motion of \(P\) is ignored. You are given that \(P\) moves in a circular path in a vertical plane containing \(O\). The angle that the rod makes with the downward vertical through \(O\) is \(\theta\) radians.

1. Determine the largest value of \(\theta\) in the subsequent motion of \(P\). In a revised model the rod is still modelled as being light and inextensible but the resistance to the motion of \(P\) is not ignored. Instead, it is modelled as causing a loss of energy of 20 J for every metre that \(P\) travels.
2. Show that according to the revised model, the maximum value of \(\theta\) in the subsequent motion of \(P\) satisfies the following equation. $$343 ( 1 + 2 \cos \theta ) = 400 \theta$$ You are given that \(\theta = 1.306\) is the solution to the above equation, correct to \(\mathbf { 4 }\) significant figures.
3. Determine the difference in the predicted maximum vertical heights attained by \(P\) using the two models. Give your answer correct to \(\mathbf { 3 }\) significant figures.
4. Suggest one further improvement that could be made to the model of the motion of \(P\).

3 A particle \(P\) of mass 5.6 kg is attached to one end of a light rod of length 2.1 m . The other end of the rod is freely hinged to a fixed point \(O\). The particle is initially at rest directly below \(O\). It is then projected horizontally with speed \(5 \mathrm {~ms} ^ { - 1 }\). In the subsequent motion, the angle between the rod and the downward vertical at \(O\) is denoted by \(\theta\) radians, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{0501e5a4-2137-4e7d-98ff-2ee81941cbf3-2_499_312_1905_244}

1. Find the speed of \(P\) when \(\theta = \frac { 1 } { 4 } \pi\).
2. Find the value of \(\theta\) when \(P\) first comes to instantaneous rest.

4 A particle \(P\) of mass 2.4 kg is moving in a straight line \(O A\) on a horizontal plane. \(P\) is acted on by a force of magnitude 30 N in the direction of motion. The distance \(O A\) is 10 m .

1. Find the work done by this force as \(P\) moves from \(O\) to \(A\). The motion of \(P\) is resisted by a constant force of magnitude \(R \mathrm {~N}\). The velocity of \(P\) increases from \(12 \mathrm {~ms} ^ { - 1 }\) at \(O\) to \(18 \mathrm {~ms} ^ { - 1 }\) at \(A\).
2. Find the value of \(R\).
3. Find the average power used in overcoming the resistance force on \(P\) as it moves from \(O\) to \(A\). When \(P\) reaches \(A\) it collides directly with a particle \(Q\) of mass 1.6 kg which was at rest at \(A\) before the collision. The impulse exerted on \(Q\) by \(P\) as a result of the collision is 17.28 Ns .
4. 1. Find the speed of \(Q\) after the collision.
   2. Hence show that the collision is inelastic.

4 A small box \(B\) of mass 4.2 kg is initially at rest at a point \(O\) on rough horizontal ground. A horizontal force of magnitude 35 N is applied to \(B\). \(B\) moves in a straight line until it reaches the point \(S\) which is 2.4 m from \(O\). At the instant that \(B\) reaches \(S\) its speed is \(4.5 \mathrm {~ms} ^ { - 1 }\).

1. 1. Find the energy lost due to the resistive forces acting on \(B\) as it moves from \(O\) to \(S\).
   2. Deduce the magnitude of the average resistive force acting on \(B\) as it moves from \(O\) to \(S\). When \(B\) reaches \(S\), the force is no longer applied. \(B\) continues to move directly up a smooth slope which is inclined at \(20 ^ { \circ }\) above the horizontal (see diagram). \includegraphics[max width=\textwidth, alt={}, center]{a65c4b75-b8b4-4a51-8abb-f857dc278271-3_275_1027_1866_244}
2. 1. State an assumption required to model the motion of \(B\) up the slope with only the information given.
   2. Using the assumption made in part (b)(i), determine the distance travelled by \(B\) up the slope until the instant when it comes to rest.

6 A smooth hemispherical shell of radius \(r \mathrm {~m}\) is held with its circular rim horizontal and uppermost. The centre of the rim is at the point \(O\) and the point on the inner surface directly below \(O\) is \(A\). A small object \(P\) of mass \(m \mathrm {~kg}\) is held at rest on the inner surface of the shell so that \(\angle \mathrm { POA } = \frac { 1 } { 3 } \pi\) radians. At the instant that \(P\) is released, an impulse is applied to \(P\) in the direction of the tangent to the surface at \(P\) in the vertical plane containing \(O , A\) and \(P\). The magnitude of the impulse is denoted by \(I\) Ns. \(P\) immediately starts to move along the surface towards \(A\) (see diagram). \(X\) is a point on the circular rim. \(P\) leaves the shell at \(X\). \includegraphics[max width=\textwidth, alt={}, center]{a65c4b75-b8b4-4a51-8abb-f857dc278271-5_512_860_829_242} In an initial model of the motion of \(P\) it is assumed that \(P\) experiences no resistance to its motion.

1. Find in terms of \(r , g , m\) and \(I\) an expression for the speed of \(P\) at the instant that it leaves the shell at \(X\).
2. Find in terms of \(r , g , m\) and \(I\) an expression for the maximum height attained by \(P\) above \(X\) after it has left the shell.
3. Find an expression for the maximum mass of \(P\) for which \(P\) still leaves the shell. In a revised model it is assumed that \(P\) experiences a resistive force of constant magnitude \(R\) while it is moving.
4. Show that, in order for \(P\) to still leave the shell at \(X\) under the revised model, $$I > \sqrt { m ^ { 2 } g r + \frac { 5 \pi m r R } { 3 } } .$$
5. Show that the inequality from part (d) is dimensionally consistent.

7 \includegraphics[max width=\textwidth, alt={}, center]{9bc86277-9e6b-41f6-a2c3-94c85e7b1360-4_330_1061_989_267} The flat surface of a smooth solid hemisphere of radius \(r\) is fixed to a horizontal plane on a planet where the acceleration due to gravity is denoted by \(\gamma\). \(O\) is the centre of the flat surface of the hemisphere. A particle \(P\) is held at a point on the surface of the hemisphere such that the angle between \(O P\) and the upward vertical through \(O\) is \(\alpha\), where \(\cos \alpha = \frac { 3 } { 4 }\). \(P\) is then released from rest. \(F\) is the point on the plane where \(P\) first hits the plane (see diagram).

1. Find an exact expression for the distance \(O F\). The acceleration due to gravity on and near the surface of the planet Earth is roughly \(6 \gamma\).
2. Explain whether \(O F\) would increase, decrease or remain unchanged if the action were repeated on the planet Earth. \section*{END OF QUESTION PAPER}

2 One end of a light elastic string of natural length 1.4 m and modulus of elasticity 20 N is attached to a small object \(B\) of mass 2.5 kg . The other end of the string is attached to a fixed point \(O\). Object \(B\) is projected vertically upwards from \(O\) with a speed of \(u \mathrm {~ms} ^ { - 1 }\).

1. State one assumption required to model the motion of \(B\). The greatest height above \(O\) achieved by \(B\) is 8.1 m .
2. Determine the value of \(u\).

2 A bungee jumper of mass 80 kg steps off a high bridge with an elastic rope attached to her ankles. She is assumed to fall vertically from rest and the air resistance she experiences is modelled as a constant force of 32 N . The rope has natural length 4 m and modulus of elasticity 470 N . By considering energy, determine the total distance she falls before first coming to instantaneous rest.

3 One end of a light inextensible string of length 0.75 m is attached to a particle \(A\) of mass 2.8 kg . The other end of the string is attached to a fixed point \(O\). \(A\) is projected horizontally with speed \(6 \mathrm {~ms} ^ { - 1 }\) from a point 0.75 m vertically above \(O\) (see Fig. 3). When \(O A\) makes an angle \(\theta\) with the upward vertical the speed of \(A\) is \(v \mathrm {~ms} ^ { - 1 }\). \begin{figure}[h] \end{figure}

1. Show that \(v ^ { 2 } = 50.7 - 14.7 \cos \theta\).
2. Given that the string breaks when the tension in it reaches 200 N , find the angle that \(O A\) turns through between the instant that \(A\) is projected and the instant that the string breaks.

1 One end of a light elastic string of natural length 0.6 m and modulus of elasticity 24 N is attached to a fixed point \(O\). The other end is attached to a particle \(P\) of mass 0.4 kg . \(O\) is a vertical distance of 1 m below a horizontal ceiling. \(P\) is held at a point 1.5 m vertically below \(O\) and released from rest (see diagram). \includegraphics[max width=\textwidth, alt={}, center]{c6445493-9802-46ca-b7eb-7738a831d9ee-2_470_371_593_255} Assuming that there is no obstruction to the motion of \(P\) as it passes \(O\), find the speed of \(P\) when it first hits the ceiling.

4 A hollow cone is fixed with its axis vertical and its vertex downwards. A small sphere \(P\) of mass \(m \mathrm {~kg}\) is moving in a horizontal circle on the inner surface of the cone. An identical sphere \(Q\) rests in equilibrium inside the cone (see diagram). \includegraphics[max width=\textwidth, alt={}, center]{c6445493-9802-46ca-b7eb-7738a831d9ee-3_586_611_404_246} The following modelling assumptions are made.

• \(P\) and \(Q\) are modelled as particles.
• The cone is modelled as smooth.
• There is no air resistance.
  1. Assuming that \(P\) moves with a constant speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), show that the total mechanical energy of \(P\) is \(\frac { 3 } { 2 } \mathrm { mv } ^ { 2 } \mathrm {~J}\) more than the total mechanical energy of \(Q\).
  2. Explain how the assumption that \(P\) and \(Q\) are both particles has been used.

In practice, \(P\) will not move indefinitely in a perfectly circular path, but will actually follow an approximately spiral path on the inside surface of the cone until eventually it collides with \(Q\).

Suggest an improvement that could be made to the model.

6 A particle \(P\) of mass 4 kg is attached to one end of a light inextensible string of length 0.8 m . The other end of the string is attached to a fixed point \(O . P\) is at rest vertically below \(O\) when it experiences a horizontal impulse of magnitude 20 Ns . In the subsequent motion the angle the string makes with the downwards vertical through \(O\) is denoted by \(\theta\) (see diagram). \includegraphics[max width=\textwidth, alt={}, center]{c6445493-9802-46ca-b7eb-7738a831d9ee-4_387_502_1434_255}

1. Find the magnitude of the acceleration of \(P\) at the first instant when \(\theta = \frac { 1 } { 3 } \pi\) radians.
2. Determine the value of \(\theta\) at which the string first becomes slack.

2 As part of a training exercise an army recruit of mass 75 kg falls a vertical distance of 5 m before landing on a mat of thickness 1.2 m . The army recruit sinks a distance of \(x \mathrm {~m}\) into the mat before instantaneously coming to rest. The mat can be modelled as a spring of natural length 1.2 m and modulus of elasticity 10800 N and the army recruit can be modelled as a particle falling vertically with an initial speed of \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Show that \(x\) satisfies the equation \(300 x ^ { 2 } - 49 x - 255 = 0\).
2. Calculate the value of \(x\).
3. Ignoring the possible effect of air resistance, make

4 A light inextensible taut rope, of length 4 m , is attached at one end \(A\) to the centre of the horizontal ceiling of a gym. The other end of the rope \(B\) is being held by a child of mass 35 kg . Initially the child is held at rest with the rope making an angle of \(60 ^ { \circ }\) to the downward vertical and it may be assumed that the child can be modelled as a particle attached to the end of the rope. The child is released at a height 5 m above the horizontal ground.

1. Show that the speed, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), of the child when the rope makes an angle \(\theta\) with the downward vertical is given by \(v ^ { 2 } = 4 g ( 2 \cos \theta - 1 )\).
2. At the instant when \(\theta = 0 ^ { \circ }\), the child lets go of the rope and moves freely under the influence of gravity only. Determine the speed and direction of the child at the moment that the child reaches the ground.
3. The child returns to the initial position and is released again from rest. Find the value of \(\theta\) when the tension in the rope is three times greater than the tension in the rope at the instant the child is released.

1 A stone, of mass 0.4 kg , is thrown vertically upwards with a speed of \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point at a height of 6 metres above ground level.

1. Calculate the initial kinetic energy of the stone.
2. 1. Show that the kinetic energy of the stone when it hits the ground is 36.3 J , correct to three significant figures.
   2. Hence find the speed at which the stone hits the ground.
   3. State one assumption that you have made.

7 A particle \(P\), of mass \(m \mathrm {~kg}\), is placed at the point \(Q\) on the top of a smooth upturned hemisphere of radius 3 metres and centre \(O\). The plane face of the hemisphere is fixed to a horizontal table. The particle is set into motion with an initial horizontal velocity of \(2 \mathrm {~ms} ^ { - 1 }\). When the particle is on the surface of the hemisphere, the angle between \(O P\) and \(O Q\) is \(\theta\) and the particle has speed \(v \mathrm {~ms} ^ { - 1 }\). \includegraphics[max width=\textwidth, alt={}, center]{6a49fdd7-f180-451c-8f37-ad764fe13dfd-4_415_1007_1573_513}

1. Show that \(v ^ { 2 } = 4 + 6 g ( 1 - \cos \theta )\).
2. Find the value of \(\theta\) when the particle leaves the hemisphere.

8 A particle, of mass 10 kg , is attached to one end of a light elastic string of natural length 0.4 metres and modulus of elasticity 100 N . The other end of the string is fixed to the point \(O\).

1. Find the length of the elastic string when the particle hangs in equilibrium directly below \(O\).
2. The particle is pulled down and held at a point \(P\), which is 1 metre vertically below \(O\). Show that the elastic potential energy of the string when the particle is in this position is 45 J .
3. The particle is released from rest at the point \(P\). In the subsequent motion, the particle has speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it is \(x\) metres below \(\boldsymbol { O }\).
   1. Show that, while the string is taut, $$v ^ { 2 } = 39.6 x - 25 x ^ { 2 } - 14.6$$
   2. Find the value of \(x\) when the particle comes to rest for the first time after being released, given that the string is still taut.