6.05e Radial/tangential acceleration

2

1. The fixed point A is vertically above the fixed point B . A light inextensible string of length 5.4 m has one end attached to A and the other end attached to B. The string passes through a small smooth ring R of mass 0.24 kg , and R is moving at constant angular speed in a horizontal circle. The circle has radius 1.6 m , and \(\mathrm { AR } = 3.4 \mathrm {~m} , \mathrm { RB } = 2.0 \mathrm {~m}\), as shown in Fig. 2 . \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{5a0df44f-f8f0-44d4-b2f6-70a5314706f9-3_565_504_447_753} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure}
   1. Find the tension in the string.
   2. Find the angular speed of R .
2. A particle P of mass 0.3 kg is joined to a fixed point O by a light inextensible string of length 1.8 m . The particle P moves without resistance in part of a vertical circle with centre O and radius 1.8 m . When OP makes an angle of \(25 ^ { \circ }\) with the downward vertical, the tension in the string is 15 N .
   1. Find the speed of P when OP makes an angle of \(25 ^ { \circ }\) with the downward vertical.
   2. Find the tension in the string when OP makes an angle of \(60 ^ { \circ }\) with the upward vertical.
   3. Find the speed of P at the instant when the string becomes slack.

2

1. A particle P of mass \(m\) is attached to a fixed point O by a light inextensible string of length \(a\). P is moving without resistance in a complete vertical circle with centre O and radius \(a\). When P is at the highest point of the circle, the tension in the string is \(T _ { 1 }\). When OP makes an angle \(\theta\) with the upward vertical, the tension in the string is \(T _ { 2 }\). Show that $$T _ { 2 } = T _ { 1 } + 3 m g ( 1 - \cos \theta ) .$$
2. The fixed point A is 1.2 m vertically above the fixed point C . A particle Q of mass 0.9 kg is joined to A , to C , and to a particle R of mass 1.5 kg , by three light inextensible strings of lengths \(1.3 \mathrm {~m} , 0.5 \mathrm {~m}\) and 1.8 m respectively. The particle Q moves in a horizontal circle with centre C , and R moves in a horizontal circle at the same constant angular speed as Q , in such a way that \(\mathrm { A } , \mathrm { C } , \mathrm { Q }\) and R are always coplanar. The string QR makes an angle of \(60 ^ { \circ }\) with the downward vertical. This situation is shown in Fig. 2. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{70a2c3ce-7bdb-4ddd-92fc-f7dcbdfdcfaf-3_579_1191_881_406} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure}
   1. Find the tensions in the strings QR and AQ .
   2. Find the angular speed of the system.
   3. Find the tension in the string CQ .

4 A particle P of mass \(m\) is attached to one end of a light inextensible string of length \(a\). The other end of the string is attached to a fixed point O . Particle P is projected so that it moves in complete vertical circles with centre O ; there is no air resistance. A and B are two points on the circle, situated on opposite sides of the vertical through O . The lines OA and OB make angles \(\alpha\) and \(\beta\) with the upward vertical as shown in Fig. 4. \begin{figure}[h] \end{figure} The speed of P at A is \(\sqrt { \frac { 17 a g } { 3 } }\). The speed of P at B is \(\sqrt { 5 a g }\) and \(\cos \beta = \frac { 2 } { 3 }\).

1. Show that \(\cos \alpha = \frac { 1 } { 3 }\). On one occasion, when P is at its lowest point and moving in a clockwise direction, it collides with a stationary particle Q . The two particles coalesce and the combined particle continues to move in the same vertical circle. When this combined particle reaches the point A , the string becomes slack.
2. Show that when the string becomes slack, the speed of the combined particle is \(\sqrt { \frac { a g } { 3 } }\). The mass of the particle Q is \(k m\).
3. Find the value of \(k\).
4. Find, in terms of \(m\) and \(g\), the instantaneous change in the tension in the string as a result of the collision.

4. On a particular day, high tide at the entrance to a harbour occurs at 11 a.m. and the water depth is 14 m . Low tide occurs \(6 \frac { 1 } { 4 }\) hours later at which time the water depth is 6 m . In a model of the situation, the water level is assumed to perform simple harmonic motion.
Using this model,

1. write down the amplitude and period of the motion. A ship needs a depth of 9 m before it can enter or leave the harbour.
2. Show that on this day a ship must enter the harbour by 2.38 p.m., correct to the nearest minute, or wait for low tide to pass.
   (6 marks)
   Given that a ship is not ready to enter the harbour until 5 p.m.,
3. find, to the nearest minute, how long the ship must wait before it can enter the harbour.

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.

2. \begin{figure}[h] \end{figure} A particle \(P\) of mass 0.5 kg is at rest at the highest point \(A\) of a smooth sphere, centre \(O\), of radius 1.25 m which is fixed to a horizontal surface. When \(P\) is slightly disturbed it slides along the surface of the sphere. Whilst \(P\) is in contact with the sphere it has speed \(v \mathrm {~ms} ^ { - 1 }\) when \(\angle A O P = \theta\) as shown in Figure 1.

1. Show that \(v ^ { 2 } = 24.5 ( 1 - \cos \theta )\).
2. Find the value of \(\cos \theta\) when \(P\) leaves the surface of the sphere.

8. A pendulum consists of a uniform rod \(P Q\), of mass \(3 m\) and length \(2 a\), which is rigidly fixed at its end \(Q\) to the centre of a uniform circular disc of mass \(m\) and radius \(a\). The rod is perpendicular to the plane of the disc. The pendulum is free to rotate about a fixed smooth horizontal axis \(L\) which passes through the end \(P\) of the rod and is perpendicular to the rod.

1. Show that the moment of inertia of the pendulum about \(L\) is \(\frac { 33 } { 4 } m a ^ { 2 }\). The pendulum is released from rest in the position where \(P Q\) makes an angle \(\alpha\) with the downward vertical. At time \(t , P Q\) makes an angle \(\theta\) with the downward vertical.
2. Show that the angular speed, \(\dot { \theta }\), of the pendulum satisfies $$\dot { \theta } ^ { 2 } = \frac { 40 g ( \cos \theta - \cos \alpha ) } { 33 a } .$$
3. Hence, or otherwise, find the angular acceleration of the pendulum. Given that \(\alpha = \frac { \pi } { 20 }\) and that \(P Q\) has length \(\frac { 8 } { 33 } \mathrm {~m}\),
4. find, to 3 significant figures, an approximate value for the angular speed of the pendulum 0.2 s after it has been released from rest. \section*{Advanced Level} \section*{Monday 25 June 2012 - Afternoon} \section*{Materials required for examination
   Mathematical Formulae (Pink)} Items included with question papers
   Nil Candidates may use any calculator allowed by the regulations of the Joint Council for Qualifications. Calculators must not have the facility for symbolic algebra manipulation, differentiation and integration, or have retrievable mathematical formulae stored in them.

4. A uniform \(\operatorname { rod } A B\), of mass \(m\) and length \(2 a\), is free to rotate in a vertical plane about a fixed smooth horizontal axis through \(A\) and perpendicular to the plane. The rod hangs in equilibrium with \(B\) below \(A\). The rod is rotated through a small angle and released from rest at time \(t = 0\).

1. Show that the motion of the rod is approximately simple harmonic.
2. Using this approximation, find the time \(t\) when the rod is first vertical after being released.
   (Total 6 marks)

7. At time \(t = 0\), a small body is projected vertically upwards. While ascending it picks up small drops of moisture from the atmosphere. The drops of moisture are at rest before they are picked up. At time \(t\), the combined body \(P\) has mass \(m\) and speed \(v\).

1. Show that, while \(P\) is moving upwards, \(m \frac { \mathrm {~d} v } { \mathrm {~d} t } + v \frac { \mathrm {~d} m } { \mathrm {~d} t } = - m g\). The initial mass of \(P\) is \(M\), and \(m = M \mathrm { e } ^ { k t }\), where \(k\) is a positive constant.
2. Show that, while \(P\) is moving upwards, \(\frac { \mathrm { d } } { \mathrm { d } t } \left( v \mathrm { e } ^ { k t } \right) = - g \mathrm { e } ^ { k t }\). Given that the initial projection speed of \(P\) is \(\frac { g } { 2 k }\),
3. find, in terms of \(M\), the mass of \(P\) when it reaches its highest point.
   (Total 15 marks)

1. A uniform circular disc, of mass \(m\) and radius \(a\), is free to rotate about a fixed smooth horizontal axis \(L\). The axis \(L\) is a tangent to the disc at the point \(A\). The centre \(O\) of the disc moves in a vertical plane that is perpendicular to \(L\).

The disc is held at rest with its plane horizontal and released.

1. Find the angular acceleration of the disc when it has turned through an angle of \(\frac { \pi } { 3 }\)
2. Find the magnitude of the component, in a direction perpendicular to the disc, of the force of the axis \(L\) acting on the disc at \(A\), when the disc has turned through an angle of \(\frac { \pi } { 3 }\)

5. \begin{figure}[h] \end{figure} A uniform piece of wire \(A B C\), of mass \(2 m\) and length \(4 a\), is bent into two straight equal portions, \(A B\) and \(B C\), which are at right angles to each other, as shown in Figure 1. The wire rotates freely in a vertical plane about a fixed smooth horizontal axis \(L\) which passes through \(A\) and is perpendicular to the plane of the wire.

1. Show that the moment of inertia of the wire about \(L\) is \(\frac { 20 m a ^ { 2 } } { 3 }\)
2. By writing down an equation of rotational motion for the wire as it rotates about \(L\), find the period of small oscillations of the wire about its position of stable equilibrium.

7. A uniform square lamina \(P Q R S\), of mass \(m\) and side \(2 a\), is free to rotate about a fixed smooth horizontal axis which passes through \(P\) and \(Q\). The lamina hangs at rest in a vertical plane with \(S R\) below \(P Q\) and is given a horizontal impulse of magnitude \(J\) at the midpoint of \(S R\). The impulse is perpendicular to \(S R\).

1. Find the initial angular speed of the lamina.
2. Find the magnitude of the angular deceleration of the lamina at the instant when the lamina has turned through \(\frac { \pi } { 6 }\) radians.
3. Find the magnitude of the component of the force exerted on the lamina by the axis, in a direction perpendicular to the lamina, at the instant when the lamina has turned through \(\frac { \pi } { 6 }\) radians. \includegraphics[max width=\textwidth, alt={}, center]{f932d7cb-1299-41d1-8248-cfbf639795ed-12_2255_50_315_1978}

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 \(D\).

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]

6. \begin{figure}[h] \end{figure} A particle \(P\) of mass \(m\) is attached to one end of a light inextensible string of length \(l\). The other end of the string is attached to a fixed point \(O\). The particle is held with the string taut and \(O P\) horizontal. The particle is then projected vertically downwards with speed \(u\), where \(u ^ { 2 } = \frac { 9 } { 5 } \mathrm { gl }\). When \(O P\) has turned through an angle \(\alpha\) and the string is still taut, the speed of \(P\) is \(v\), as shown in Figure 5. At this instant the tension in the string is \(T\).

1. Show that \(T = 3 m g \sin \alpha + \frac { 9 } { 5 } m g\)
2. Find, in terms of \(g\) and \(l\), the speed of \(P\) at the instant when the string goes slack.
3. Find, in terms of \(l\), the greatest vertical height reached by \(P\) above the level of \(O\).

1. A light inextensible string of length \(a\) has one end attached to a fixed point \(O\). The other end of the string is attached to a small stone of mass \(m\). The stone is held with the string taut and horizontal. The stone is then projected vertically upwards with speed \(U\).

The stone is modelled as a particle and air resistance is modelled as being negligible.
Assuming that the string does not break, use the model to

1. find the least value of \(U\) so that the stone will move in complete vertical circles. The string will break if the tension in it is equal to \(\frac { 11 m g } { 2 }\) Given that \(U = 2 \sqrt { a g }\), use the model to
2. find the total angle that the string has turned through, from when the stone is projected vertically upwards, to when the string breaks,
3. find the magnitude of the acceleration of the stone at the instant just before the string breaks.

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.

4. \begin{figure}[h] \end{figure} A smooth hemisphere of radius \(a\) is fixed on a horizontal surface with its plane face in contact with the surface. The centre of the plane face of the hemisphere is \(O\). A particle \(P\) of mass \(M\) is disturbed from rest at the highest point of the hemisphere.
When \(P\) is still on the surface of the hemisphere and the radius from \(O\) to \(P\) is at an angle \(\theta\) to the vertical,

• the speed of \(P\) is \(v\)
• the normal reaction between the hemisphere and the particle is \(R\), as shown in Figure 2.
  1. Show that \(\mathrm { R } = \mathrm { Mg } ( 3 \cos \theta - 2 )\)
  2. Find, in terms of \(a\) and \(g\), the speed of the particle at the instant when the particle leaves the surface of the hemisphere.

7. \begin{figure}[h] \end{figure} A smooth solid hemisphere has radius \(r\) and the centre of its plane face is \(O\).
The hemisphere is fixed with its plane face in contact with horizontal ground, as shown in Figure 6.
A small stone is at the point \(A\), the highest point on the surface of the hemisphere. The stone is projected horizontally from \(A\) with speed \(U\).
The stone is still in contact with the hemisphere at the point \(B\), where \(O B\) makes an angle \(\theta\) with the upward vertical.
The speed of the stone at the instant it reaches \(B\) is \(v\).
The stone is modelled as a particle \(P\) and air resistance is modelled as being negligible.

1. Use the model to find \(v ^ { 2 }\) in terms of \(U , r , g\) and \(\theta\) When \(P\) leaves the surface of the hemisphere, the speed of \(P\) is \(W\).
   Given that \(U = \sqrt { \frac { 2 r g } { 3 } }\)
2. show that \(W ^ { 2 } = \frac { 8 } { 9 } r g\) After leaving the surface of the hemisphere, \(P\) moves freely under gravity until it hits the ground.
3. Find the speed of \(P\) as it hits the ground, giving your answer in terms of \(r\) and \(g\). At the instant when \(P\) hits the ground it is travelling at \(\alpha ^ { \circ }\) to the horizontal.
4. Find the value of \(\alpha\).

1. A small bead \(B\) of mass \(m\) is threaded on a circular hoop.

The hoop has centre \(O\) and radius \(a\) and is fixed in a vertical plane.
The bead is projected with speed \(\sqrt { \frac { 7 } { 2 } g a }\) from the lowest point of the hoop.
The hoop is modelled as being smooth.
When the angle between \(O B\) and the downward vertical is \(\theta\), the speed of \(B\) is \(v\).

1. Show that \(v ^ { 2 } = g a \left( \frac { 3 } { 2 } + 2 \cos \theta \right)\)
2. Find the size of \(\theta\) at the instant when the contact force between \(B\) and the hoop is first zero.
3. Give a reason why your answer to part (b) is not likely to be the actual value of \(\theta\).
4. Find the magnitude and direction of the acceleration of \(B\) at the instant when \(B\) is first at instantaneous rest.

6 A fairground game involves a player kicking a ball, \(B\), from rest so as to project it with a horizontal velocity of magnitude \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The ball is attached to one end of a light rod of length \(l \mathrm {~m}\). The other end of the rod is smoothly hinged at a fixed point \(O\) so that \(B\) can only move in the vertical plane which contains \(O\), a fixed barrier and a bell which is fixed \(l \mathrm {~m}\) vertically above \(O\). Initially \(B\) is vertically below \(O\). The barrier is positioned so that when \(B\) collides directly with the barrier, \(O B\) makes an angle \(\theta\) with the downwards vertical through \(O\) (see diagram). \includegraphics[max width=\textwidth, alt={}, center]{bf86ac88-0fd1-4d49-a705-9b8d06fbac2a-4_643_659_584_724} The coefficient of restitution between \(B\) and the barrier is \(e . B\) rebounds from the barrier, passes through its original position and continues on a circular path towards the bell. The bell will only ring if the ball strikes it with a speed of at least \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The player wins the game if the player causes the bell to ring having kicked \(B\) so that it first collides with the barrier. You may assume that \(B\) and the bell are small and that the barrier has negligible thickness. Show that, whatever the position of the barrier, the player cannot win the game if \(u ^ { 2 } < 4 g l + \frac { V ^ { 2 } } { e ^ { 2 } }\). \section*{END OF QUESTION PAPER}

5 One end of a light inextensible string of length 0.8 m is attached to a fixed point, \(O\). The other end is attached to a particle \(P\) of mass \(1.2 \mathrm {~kg} . P\) hangs in equilibrium at a distance of 1.5 m above a horizontal plane. The point on the plane directly below \(O\) is \(F\). \(P\) is projected horizontally with speed \(3.5 \mathrm {~ms} ^ { - 1 }\). The string breaks when \(O P\) makes an angle of \(\frac { 1 } { 3 } \pi\) radians with the downwards vertical through \(O\) (see diagram). \includegraphics[max width=\textwidth, alt={}, center]{493f11f4-e25c-4eeb-a0ab-20ec6d7a7a7d-3_776_910_1242_244}

1. Find the magnitude of the tension in the string at the instant before the string breaks.
2. Find the distance between \(F\) and the point where \(P\) first hits the plane.