6.05d Variable speed circles: energy methods

6 Simon, a small child of mass 22 kg , is on a swing. He is swinging freely through an angle of \(18 ^ { \circ }\) on both sides of the vertical. Model Simon as a particle, \(P\), of mass 22 kg , attached to a fixed point, \(Q\), by a light inextensible rope of length 2.4 m . \includegraphics[max width=\textwidth, alt={}, center]{088327c1-acd3-486d-b76f-1fe2560ffaff-5_700_310_466_849}

1. Find Simon's maximum speed as he swings.
2. Calculate the tension in the rope when Simon's speed is a maximum.

8 A bead, of mass \(m\), moves on a smooth circular ring, of radius \(a\) and centre \(O\), which is fixed in a vertical plane. At \(P\), the highest point on the ring, the speed of the bead is \(2 u\); at \(Q\), the lowest point on the ring, the speed of the bead is \(5 u\).

1. Show that \(u = \sqrt { \frac { 4 a g } { 21 } }\).
   (4 marks)
2. \(\quad S\) is a point on the ring so that angle \(P O S\) is \(60 ^ { \circ }\), as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{5b1c9e8d-459a-474c-bd29-6dadff40de14-4_600_540_657_760} Find, in terms of \(m\) and \(g\), the magnitude of the reaction of the ring on the bead when the bead is at \(S\).

7 \includegraphics[max width=\textwidth, alt={}, center]{5bfd0285-71cb-4dcb-8545-a379653f9a3e-4_529_403_264_829} A small smooth ring \(P\) of mass 0.4 kg is threaded onto a light inextensible string fixed at \(A\) and \(B\) as shown in the diagram, with \(A\) vertically above \(B\). The string is inclined to the vertical at angles of \(30 ^ { \circ }\) and \(45 ^ { \circ }\) at \(A\) and \(B\) respectively. \(P\) moves in a horizontal circle of radius 0.5 m about a point \(C\) vertically below \(B\).

1. Calculate the tension in the string.
2. Calculate the speed of \(P\). The end of the string at \(B\) is moved so both ends of the string are now fixed at \(A\).
3. Show that, when the string is taut, \(A P\) is now 0.854 m correct to 3 significant figures. \(P\) moves in a horizontal circle with angular speed \(3.46 \mathrm { rad } \mathrm { s } ^ { - 1 }\).
4. Find the tension in the string and the angle that the string now makes with the vertical.

7 A simple pendulum consists of a light inextensible string of length 0.8 m and a particle \(P\) of mass \(m \mathrm {~kg}\). The pendulum is hanging vertically at rest from a fixed point \(O\) when \(P\) is given a horizontal velocity of \(0.3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Show that, in the subsequent motion, the maximum angle between the string and the downward vertical is 0.107 radians, correct to 3 significant figures.
2. Show that the motion may be modelled as simple harmonic motion, and find the period of this motion.
3. Find the time after the start of the motion when the velocity of the particle is first \(- 0.2 \mathrm {~ms} ^ { - 1 }\) and find the angular displacement of \(O P\) from the downward vertical at this time.

5 \includegraphics[max width=\textwidth, alt={}, center]{3e8248ca-74f1-443f-a5db-d7da532d2815-3_449_442_1281_794} A fixed smooth sphere of radius 0.6 m has centre \(O\) and highest point \(T\). A particle of mass \(m \mathrm {~kg}\) is released from rest at a point \(A\) on the sphere, such that angle \(T O A\) is \(\frac { \pi } { 6 }\) radians. The particle leaves the surface of the sphere at \(B\) (see diagram).

1. Show that \(\cos T O B = \frac { \sqrt { 3 } } { 3 }\).
2. Find the speed of the particle at \(B\).
3. Find the transverse acceleration of the particle at \(B\).

6 \includegraphics[max width=\textwidth, alt={}, center]{3243c326-a51c-462f-a57c-a150d0044ea9-4_547_515_267_772} A hollow cylinder is fixed with its axis horizontal. \(O\) is the centre of a vertical cross-section of the cylinder and \(D\) is the highest point on the cross-section. \(A\) and \(C\) are points on the circumference of the cross-section such that \(A O\) and \(C O\) are both inclined at an angle of \(30 ^ { \circ }\) below the horizontal diameter through \(O\). The inner surface of the cylinder is smooth and has radius 0.8 m (see diagram). A particle \(P\), of mass \(m \mathrm {~kg}\), and a particle \(Q\), of mass \(5 m \mathrm {~kg}\), are simultaneously released from rest from \(A\) and \(C\), respectively, inside the cylinder. \(P\) and \(Q\) collide; the coefficient of restitution between them is 0.95 .

1. Show that, immediately after the collision, \(P\) moves with speed \(6.3 \mathrm {~ms} ^ { - 1 }\), and find the speed and direction of motion of \(Q\).
2. Find, in terms of \(m\), an expression for the normal reaction acting on \(P\) when it subsequently passes through \(D\).

5 \includegraphics[max width=\textwidth, alt={}, center]{bfa6d51d-0992-4f43-adab-77ce893c1ca9-3_576_535_258_804} A particle \(P\) of mass 0.3 kg is moving in a vertical circle. It is attached to the fixed point \(O\) at the centre of the circle by a light inextensible string of length 1.5 m . When the string makes an angle of \(40 ^ { \circ }\) with the downward vertical, the speed of \(P\) is \(6.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) (see diagram). Air resistance may be neglected.

1. Find the radial and transverse components of the acceleration of \(P\) at this instant. In the subsequent motion, with the string still taut and making an angle \(\theta ^ { \circ }\) with the downward vertical, the speed of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\)
2. Use conservation of energy to show that \(v ^ { 2 } \approx 19.7 + 29.4 \cos \theta ^ { \circ }\).
3. Find the tension in the string in terms of \(\theta\).
4. Find the value of \(v\) at the instant when the string becomes slack. \includegraphics[max width=\textwidth, alt={}, center]{bfa6d51d-0992-4f43-adab-77ce893c1ca9-3_574_842_1640_664} A step-ladder is modelled as two uniform rods \(A B\) and \(A C\), freely jointed at \(A\). The rods are in equilibrium in a vertical plane with \(B\) and \(C\) in contact with a rough horizontal surface. The rods have equal lengths; \(A B\) has weight 150 N and \(A C\) has weight 270 N . The point \(A\) is 2.5 m vertically above the surface, and \(B C = 1.6 \mathrm {~m}\) (see diagram).
5. Find the horizontal and vertical components of the force acting on \(A C\) at \(A\).
6. The coefficient of friction has the same value \(\mu\) at \(B\) and at \(C\), and the step-ladder is on the point of slipping. Giving a reason, state whether the equilibrium is limiting at \(B\) or at \(C\), and find \(\mu\). \includegraphics[max width=\textwidth, alt={}, center]{bfa6d51d-0992-4f43-adab-77ce893c1ca9-4_648_227_269_982} Two points \(A\) and \(B\) lie on a vertical line with \(A\) at a distance 2.6 m above \(B\). A particle \(P\) of mass 10 kg is joined to \(A\) by an elastic string and to \(B\) by another elastic string (see diagram). Each string has natural length 0.8 m and modulus of elasticity 196 N . The strings are light and air resistance may be neglected.
7. Verify that \(P\) is in equilibrium when \(P\) is vertically below \(A\) and the length of the string \(P A\) is 1.5 m . The particle is set in motion along the line \(A B\) with both strings remaining taut. The displacement of \(P\) below the equilibrium position is denoted by \(x\) metres.
8. Show that the tension in the string \(P A\) is \(245 ( 0.7 + x )\) newtons, and the tension in the string \(P B\) is \(245 ( 0.3 - x )\) newtons.
9. Show that the motion of \(P\) is simple harmonic.
10. Given that the amplitude of the motion is 0.25 m , find the proportion of time for which \(P\) is above the mid-point of \(A B\).

2

1. A moon of mass \(7.5 \times 10 ^ { 22 } \mathrm {~kg}\) moves round a planet in a circular path of radius \(3.8 \times 10 ^ { 8 } \mathrm {~m}\), completing one orbit in a time of \(2.4 \times 10 ^ { 6 } \mathrm {~s}\). Find the force acting on the moon.
2. Fig. 2 shows a fixed solid sphere with centre O and radius 4 m . Its surface is smooth. The point A on the surface of the sphere is 3.5 m vertically above the level of O . A particle P of mass 0.2 kg is placed on the surface at A and is released from rest. In the subsequent motion, when OP makes an angle \(\theta\) with the horizontal and P is still on the surface of the sphere, the speed of P is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the normal reaction acting on P is \(R \mathrm {~N}\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{e0e5580a-e1f0-46f8-9304-2a96533af186-03_746_734_705_662} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure}
   1. Express \(v ^ { 2 }\) in terms of \(\theta\).
   2. Show that \(R = 5.88 \sin \theta - 3.43\).
   3. Find the radial and tangential components of the acceleration of P when \(\theta = 40 ^ { \circ }\).
   4. Find the value of \(\theta\) at the instant when P leaves the surface of the sphere.

2

1. A moon of mass \(7.5 \times 10 ^ { 22 } \mathrm {~kg}\) moves round a planet in a circular path of radius \(3.8 \times 10 ^ { 8 } \mathrm {~m}\), completing one orbit in a time of \(2.4 \times 10 ^ { 6 } \mathrm {~s}\). Find the force acting on the moon.
2. Fig. 2 shows a fixed solid sphere with centre O and radius 4 m . Its surface is smooth. The point A on the surface of the sphere is 3.5 m vertically above the level of O . A particle P of mass 0.2 kg is placed on the surface at A and is released from rest. In the subsequent motion, when OP makes an angle \(\theta\) with the horizontal and P is still on the surface of the sphere, the speed of P is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the normal reaction acting on P is \(R \mathrm {~N}\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{b7f8bdfd-33dc-4453-8f3a-ddd24be17372-3_746_734_705_662} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure}
   1. Express \(v ^ { 2 }\) in terms of \(\theta\).
   2. Show that \(R = 5.88 \sin \theta - 3.43\).
   3. Find the radial and tangential components of the acceleration of P when \(\theta = 40 ^ { \circ }\).
   4. Find the value of \(\theta\) at the instant when P leaves the surface of the sphere.

2

1. A light inextensible string has length 1.8 m . One end of the string is attached to a fixed point O , and the other end is attached to a particle of mass 5 kg . The particle moves in a complete vertical circle with centre O , so that the string remains taut throughout the motion. Air resistance may be neglected.
   1. Show that, at the highest point of the circle, the speed of the particle is at least \(4.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
   2. Find the least possible tension in the string when the particle is at the lowest point of the circle.
2. Fig. 2 shows a hollow cone mounted with its axis of symmetry vertical and its vertex V pointing downwards. The cone rotates about its axis with a constant angular speed of \(\omega \mathrm { rad } \mathrm { s } ^ { - 1 }\). A particle P of mass 0.02 kg is in contact with the rough inside surface of the cone, and does not slip. The particle P moves in a horizontal circle of radius 0.32 m . The angle between VP and the vertical is \(\theta\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{b209dbe7-769c-4301-a2f3-108c27c8cefb-3_588_510_1046_772} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure} In the case when \(\omega = 8.75\), there is no frictional force acting on P .
   1. Show that \(\tan \theta = 0.4\). Now consider the case when \(\omega\) takes a constant value greater than 8.75.
   2. Draw a diagram showing the forces acting on P .
   3. You are given that the coefficient of friction between P and the surface is 0.11 . Find the maximum possible value of \(\omega\) for which the particle does not slip.

2

1. A small ball of mass 0.01 kg is moving in a vertical circle of radius 0.55 m on the smooth inside surface of a fixed sphere also of radius 0.55 m . When the ball is at the highest point of the circle, the normal reaction between the surface and the ball is 0.1 N . Modelling the ball as a particle and neglecting air resistance, find
   1. the speed of the ball when it is at the highest point of the circle,
   2. the normal reaction between the surface and the ball when the vertical height of the ball above the lowest point of the circle is 0.15 m .
2. A small object Q of mass 0.8 kg moves in a circular path, with centre O and radius \(r\) metres, on a smooth horizontal surface. A light elastic string, with natural length 2 m and modulus of elasticity 160 N , has one end attached to Q and the other end attached to O . The object Q has a constant angular speed of \(\omega\) rad s \(^ { - 1 }\).
   1. Show that \(\omega ^ { 2 } = \frac { 100 ( r - 2 ) } { r }\) and deduce that \(\omega < 10\).
   2. Find expressions, in terms of \(r\) only, for the elastic energy stored in the string, and for the kinetic energy of Q . Show that the kinetic energy of Q is greater than the elastic energy stored in the string.
   3. Given that the angular speed of Q is \(6 \mathrm { rad } \mathrm { s } ^ { - 1 }\), find the tension in the string.

2

1. Fig. 2 shows a light inextensible string of length 3.3 m passing through a small smooth ring R of mass 0.27 kg . The ends of the string are attached to fixed points A and B , where A is vertically above \(B\). The ring \(R\) is moving with constant speed in a horizontal circle of radius \(1.2 \mathrm {~m} , \mathrm { AR } = 2.0 \mathrm {~m}\) and \(\mathrm { BR } = 1.3 \mathrm {~m}\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{b8573ee2-771c-4a93-88d9-346a9da94494-3_570_659_493_781} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure}
   1. Show that the tension in the string is 6.37 N .
   2. Find the speed of R .
2. One end of a light inextensible string of length 1.25 m is attached to a fixed point O . The other end is attached to a particle P of mass 0.2 kg . The particle P is moving in a vertical circle with centre O and radius 1.25 m , and when P is at the highest point of the circle there is no tension in the string.
   1. Show that when P is at the highest point its speed is \(3.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). For the instant when the string OP makes an angle of \(60 ^ { \circ }\) with the upward vertical, find
   2. the radial and tangential components of the acceleration of P ,
   3. the tension in the string.

3 A particle P of mass 0.6 kg is connected to a fixed point O by a light inextensible string of length 1.25 m . When it is 1.25 m vertically below \(\mathrm { O } , \mathrm { P }\) is set in motion with horizontal velocity \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and then moves in part of a vertical circle with centre O and radius 1.25 m . When OP makes an angle \(\theta\) with the downward vertical, the speed of P is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), as shown in Fig. 3.1. \begin{figure}[h] \end{figure}

1. Show that \(v ^ { 2 } = 11.5 + 24.5 \cos \theta\).
2. Find the tension in the string in terms of \(\theta\).
3. Find the speed of P at the instant when the string becomes slack. A second light inextensible string, of length 0.35 m , is attached to P , and the other end of this string is attached to a point C which is 1.2 m vertically below O . The particle P now moves in a horizontal circle with centre C and radius 0.35 m , as shown in Fig. 3.2. The speed of P is \(1.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{023afdfb-21b6-40fe-9a09-e6769667ee7b-3_518_488_1701_826} \captionsetup{labelformat=empty} \caption{Fig. 3.2}
   \end{figure}
4. Find the tension in the string OP and the tension in the string CP.

2

1. A particle P , of mass 48 kg , is moving in a horizontal circle of radius 8.4 m at a constant speed of \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\), in contact with a smooth horizontal surface. A light inextensible rope of length 30 m connects P to a fixed point A which is vertically above the centre C of the circle, as shown in Fig. 2.1. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{f2dd5719-bef3-45f2-afd2-c481e6a4b129-3_526_490_482_870} \captionsetup{labelformat=empty} \caption{Fig. 2.1}
   \end{figure}
   1. Given that \(V = 3.5\), find the tension in the rope and the normal reaction of the surface on P .
   2. Calculate the value of \(V\) for which the normal reaction is zero.
2. The particle P , of mass 48 kg , is now placed on the highest point of a fixed solid sphere with centre O and radius 2.5 m . The surface of the sphere is smooth. The particle P is given an initial horizontal velocity of \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\), and it then moves in part of a vertical circle with centre O and radius 2.5 m . When OP makes an angle \(\theta\) with the upward vertical and P is still in contact with the surface of the sphere, P has speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the normal reaction of the sphere on P is \(R \mathrm {~N}\), as shown in Fig. 2.2. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{f2dd5719-bef3-45f2-afd2-c481e6a4b129-3_590_617_1706_804} \captionsetup{labelformat=empty} \caption{Fig. 2.2}
   \end{figure}
   1. Show that \(v ^ { 2 } = u ^ { 2 } + 49 - 49 \cos \theta\).
   2. Find an expression for \(R\) in terms of \(u\) and \(v\).
   3. Given that P loses contact with the surface of the sphere at the instant when its speed is \(4.15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), find the value of \(u\).

2 A light inextensible string of length 5 m has one end attached to a fixed point A and the other end attached to a particle P of mass 0.72 kg . At first, P is moving in a vertical circle with centre A and radius 5 m . When P is at the highest point of the circle it has speed \(10 \mathrm {~ms} ^ { - 1 }\).

1. Find the tension in the string when the speed of P is \(15 \mathrm {~ms} ^ { - 1 }\). The particle P now moves at constant speed in a horizontal circle with radius 1.4 m and centre at the point C which is 4.8 m vertically below A .
2. Find the tension in the string.
3. Find the time taken for P to make one complete revolution. Another light inextensible string, also of length 5 m , now has one end attached to P and the other end attached to the fixed point B which is 9.6 m vertically below A . The particle P then moves with constant speed \(7 \mathrm {~ms} ^ { - 1 }\) in the circle with centre C and radius 1.4 m , as shown in Fig. 2. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{86d79489-aec1-4c94-bef6-45b007f818a0-3_693_465_1078_817} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure}
4. Find the tension in the string PA and the tension in the string PB .

3. \begin{figure}[h] \end{figure} A particle of mass \(m\) is suspended at a point \(A\) vertically below a fixed point \(O\) by a light inextensible string of length \(a\) as shown in Figure 2. The particle is given a horizontal velocity \(u\) and subsequently moves along a circular arc until it reaches the point \(B\) where the string becomes slack. Given that the point \(B\) is at a height \(\frac { 1 } { 2 } a\) above the level of \(O\),

1. show that \(\angle B O A = 120 ^ { \circ }\),
2. show that \(u ^ { 2 } = \frac { 7 } { 2 } g a\).

7. \begin{figure}[h] \end{figure} Figure 3 shows a vertical cross-section through part of a ski slope consisting of a horizontal section \(A B\) followed by a downhill section \(B C\). The point \(O\) is on the same horizontal level as \(C\) and \(B C\) is a circular arc of radius 30 m and centre \(O\), such that \(\angle B O C = 90 ^ { \circ }\). A skier of mass 60 kg is skiing at \(12 \mathrm {~ms} ^ { - 1 }\) along \(A B\).

1. Assuming that friction and air resistance may be neglected, find the magnitude of the loss in reaction between the skier and the surface at \(B\).
   (4 marks)
   The skier subsequently leaves the slope at the point \(P\).
2. Find, correct to 3 significant figures, the speed at which the skier leaves the slope.
3. Find, correct to 3 significant figures, the speed of the skier immediately before hitting the ground again at the point \(D\) which is on the same horizontal level as \(C\).

2. A small bead \(P\) is threaded onto a smooth circular wire of radius 0.8 m and centre \(O\) which is fixed in a vertical plane. The bead is projected from the point vertically below \(O\) with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and moves in complete circles about \(O\).

1. Suggest a suitable model for the bead.
2. Given that the minimum speed of \(P\) is \(60 \%\) of its maximum speed, use the principle of conservation of energy to show that \(u = 7\).
   (6 marks)

7. A particle of mass 0.5 kg is hanging vertically at one end of a light inextensible string of length 0.6 m . The other end of the string is attached to a fixed point. The particle is given an initial horizontal speed of \(u \mathrm {~ms} ^ { - 1 }\).

1. Show that the particle will perform complete circles if \(u \geq \sqrt { 3 g }\). Given that \(u = 5\),
2. find, correct to the nearest degree, the angle through which the string turns before it becomes slack,
3. find, correct to the nearest centimetre, the greatest height the particle reaches above its position when the string becomes slack.

6 A smooth solid hemisphere of radius \(a\) is fixed with its plane face in contact with a horizontal surface.
The highest point on the hemisphere is H , and the centre of its base is O . A particle of mass \(m\) is held at a point S on the surface of the hemisphere such that angle HOS is \(30 ^ { \circ }\), as shown in Fig. 6. The particle is projected from S with speed \(0.8 \sqrt { a g }\) along the surface of the hemisphere towards H . \begin{figure}[h] \end{figure}

1. Show that the particle passes through H without leaving the surface of the hemisphere. After passing through H , the particle passes through a point Q on the surface of the hemisphere, where angle \(\mathrm { HOQ } = \theta ^ { \circ }\).
2. State, in terms of \(g\) and \(\theta\), the tangential component of the acceleration of the particle when it is at Q . The particle loses contact with the hemisphere at Q and subsequently lands on the horizontal surface at a point L .
3. Find the value of \(\cos \theta\) correct to 3 significant figures.
4. Show that \(\mathrm { OL } = k a\), where \(k\) is to be found correct to 3 significant figures.

6. The diagram shows a rollercoaster at an amusement park where a car is projected from a launch point \(O\) so that it performs a loop before instantaneously coming to rest at point \(C\). The car then performs the same journey in reverse. \includegraphics[max width=\textwidth, alt={}, center]{b430aa50-27e3-46f7-afef-7b8e75d46e1f-5_677_1733_552_166} The loop section is modelled by considering the track to be a vertical circle of radius 10 m and the car as a particle of mass \(m\) kg moving on the inside surface of the circular loop. You may assume that the track is smooth. At point \(A\), which is the lowest point of the circle, the car has velocity \(u \mathrm {~ms} ^ { - 1 }\) such that \(u ^ { 2 } = 60 g\). When the car is at point \(B\) the radius makes an angle \(\theta\) with the downward vertical.

1. Find, in terms of \(\theta\) and \(g\), an expression for \(v ^ { 2 }\), where \(v \mathrm {~ms} ^ { - 1 }\) is the speed of the car at \(B\).
2. Show that \(R \mathrm {~N}\), the reaction of the track on the car at \(B\), is given by $$R = m g ( 4 + 3 \cos \theta ) .$$
3. Explain why the expression for \(R\) in part (b) shows that the car will perform a complete loop.
4. This model predicts that the car will stop at \(C\) at a vertical height of 30 m above \(A\). However, after the car has completed the loop, the track becomes rough and the car only reaches a point \(D\) at a vertical height of 28 m above \(A\). The resistance to motion of the car beyond the loop is of constant magnitude \(\frac { m g } { 32 } \mathrm {~N}\). Calculate the length of the rough track between \(A\) and \(D\).

7. One end of a light rod of length \(\frac { 5 } { 7 } \mathrm {~m}\) is attached to a fixed point \(O\) and the other end is attached to a particle \(P\), of mass \(m \mathrm {~kg}\). The particle \(P\) is projected from the point \(A\), which is vertically below \(O\), with a horizontal speed of \(u \mathrm {~ms} ^ { - 1 }\) so that it moves in a vertical circle with centre \(O\). When the rod \(O P\) is inclined at an angle \(\theta\) to the downward vertical, the speed of \(P\) is \(v \mathrm {~ms} ^ { - 1 }\) and the tension in the rod is \(T \mathrm {~N}\). \includegraphics[max width=\textwidth, alt={}, center]{ae23a093-1419-4be4-8285-951650dc5a35-16_629_593_646_735}

1. Show that $$v ^ { 2 } = u ^ { 2 } - 14 + 14 \cos \theta$$
2. Hence determine the least possible value of \(u ^ { 2 }\) for the particle to reach the highest point of the circle.
3. Given that \(u ^ { 2 } = 32 \cdot 2\),
   1. find, in terms of \(m\) and \(\theta\), an expression for \(T\),
   2. calculate the range of values of \(\theta\) such that the rod is exerting a thrust.
      State whether your answer to (c)(ii) would be different if the mass of the particle was reduced. Give a reason for your answer. Additional page, if required. Write the question number(s) in the left-hand margin. only

8 The diagram shows part of a water park slide, \(A B C\).
The slide is in the shape of two circular arcs, \(A B\) and \(B C\), each of radius \(r\).
The point \(A\) is at a height of \(\frac { r } { 4 }\) above \(B\).
The circular \(\operatorname { arc } B C\) has centre \(O\) and \(B\) is vertically above \(O\).
These points are joined as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{4fdb2637-6368-422c-99da-85b80efe31c5-12_590_1173_756_443} A child starts from rest at \(A\), moves along the slide past the point \(B\) and then loses contact with the slide at a point \(D\). The angle between the vertical, \(O B\), and \(O D\) is \(\theta\) Assume that the slide is smooth. 8

1. Show that the speed \(v\) of the child at \(D\) is given by \(v = \sqrt { \frac { g r } { 2 } ( 5 - 4 \cos \theta ) }\), where \(g\) is the acceleration due to gravity. 8
2. Find \(\theta\), giving your answer to the nearest degree.
   8
3. A refined model takes into account air resistance. Explain how taking air resistance into account would affect your answer to part (b).
   [0pt] [2 marks]
   8
4. In reality the slide is not smooth. It has a surface with the same coefficient of friction between the slide and the child for its entire length. Explain why the frictional force experienced by the child is not constant.
   [0pt] [1 mark]

2. \begin{figure}[h] \end{figure} A thin hollow hemisphere, with centre \(O\) and radius \(a\), is fixed with its axis vertical, as shown in Figure 2. A small ball \(B\) of mass \(m\) moves in a horizontal circle on the inner surface of the hemisphere. The circle has centre \(C\) and radius \(r\). The point \(C\) is vertically below \(O\) such that \(O C = h\). The ball moves with constant angular speed \(\omega\) The inner surface of the hemisphere is modelled as being smooth and \(B\) is modelled as a particle. Air resistance is modelled as being negligible.

1. Show that \(\omega ^ { 2 } = \frac { g } { h }\) Given that the magnitude of the normal reaction between \(B\) and the surface of the hemisphere is \(3 m g\)
2. find \(\omega\) in terms of \(g\) and \(a\).
3. State how, apart from ignoring air resistance, you have used the fact that \(B\) is modelled as a particle.

7. \begin{figure}[h] \end{figure} A package \(P\) of mass \(m\) is attached to one end of a string of length \(\frac { 2 a } { 5 }\). The other end of the string is attached to a fixed point \(O\). The package hangs at rest vertically below \(O\) with the string taut and is then projected horizontally with speed \(u\), as shown in Figure 5. When \(O P\) has turned through an angle \(\theta\) and the string is still taut, the tension in the string is \(T\) The package is modelled as a particle and the string as being light and inextensible.

1. Show that \(T = 3 m g \cos \theta - 2 m g + \frac { 5 m u ^ { 2 } } { 2 a }\) Given that \(P\) moves in a complete vertical circle with centre \(O\)
2. find, in terms of \(a\) and \(g\), the minimum possible value of \(u\) Given that \(u = 2 \sqrt { a g }\)
3. find, in terms of \(g\), the magnitude of the acceleration of \(P\) at the instant when \(O P\) is horizontal.
4. Apart from including air resistance, suggest one way in which the model could be refined to make it more realistic.