Circular Motion 2

1. Show that \(v ^ { 2 } = 9 + 9.8 \sin \theta\).
2. Find, in terms of \(\theta\), the radial and tangential components of the acceleration of \(P\).
3. Show that the tension in the string is \(( 3.6 + 5.88 \sin \theta ) \mathrm { N }\) and hence find the value of \(\theta\) at the instant when the string becomes slack, giving your answer correct to 1 decimal place.

6
\includegraphics[max width=\textwidth, alt={}, center]{5bb3bd29-a2eb-4124-802c-fb17b68c50e4-3_598_839_1480_706} One end of a light inextensible string of length 0.5 m is attached to a fixed point \(O\). A particle \(P\) of mass 0.3 kg is attached to the other end of the string. With the string taut and at an angle of \(60 ^ { \circ }\) to the upward vertical, \(P\) is projected with speed \(2 \mathrm {~ms} ^ { - 1 }\) (see diagram). \(P\) begins to move without air resistance in a vertical circle with centre \(O\). When the string makes an angle \(\theta\) with the upward vertical, the speed of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Show that \(v ^ { 2 } = 8.9 - 9.8 \cos \theta\).
2. Find the tension in the string in terms of \(\theta\).
3. \(P\) does not move in a complete circle. Calculate the angle through which \(O P\) turns before \(P\) leaves the circular path.

1 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 \(O\). The string is held taut with \(O P\) making an angle \(\alpha\) with the downward vertical, where \(\cos \alpha = \frac { 2 } { 3 }\). The particle \(P\) is projected perpendicular to \(O P\) in an upwards direction with speed \(\sqrt { 3 a g }\). It then starts to move along a circular path in a vertical plane. Find the cosine of the angle between the string and the upward vertical when the string first becomes slack.
\includegraphics[max width=\textwidth, alt={}, center]{9b3f3add-17fd-4597-bd5d-27e3abb527be-03_671_455_255_845} A uniform lamina is in the form of a triangle \(A B C\) in which angle \(B\) is a right angle, \(\mathrm { AB } = 9 \mathrm { a }\) and \(\mathrm { BC } = 6 \mathrm { a }\). The point \(D\) is on \(B C\) such that \(\mathrm { BD } = \mathrm { x }\) (see diagram). The region \(A B D\) is removed from the lamina. The resulting shape \(A D C\) is placed with the edge \(D C\) on a horizontal surface and the plane \(A D C\) is vertical. Find the set of values of \(x\), in terms of \(a\), for which the shape is in equilibrium.

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 \(B\).

5. A smooth solid sphere, with centre \(O\) and radius \(a\), is fixed to the upper surface of a horizontal table. A particle \(P\) is placed on the surface of the sphere at a point \(A\), where \(O A\) makes an angle \(\alpha\) with the upward vertical, and \(0 < \alpha < \frac { \pi } { 2 }\). The particle is released from rest. When \(O P\) makes an angle \(\theta\) with the upward vertical, and \(P\) is still on the surface of the sphere, the speed of \(P\) is \(v\).

1. Show that \(v ^ { 2 } = 2 g a ( \cos \alpha - \cos \theta )\). Given that \(\cos \alpha = \frac { 3 } { 4 }\), find
2. the value of \(\theta\) when \(P\) loses contact with the sphere,
3. the speed of \(P\) as it hits the table.
   (Total 13 marks)

7. \begin{figure}[h] \end{figure} A particle is projected from the highest point \(A\) on the outer surface of a fixed smooth sphere of radius \(a\) and centre \(O\). The lowest point \(B\) of the sphere is fixed to a horizontal plane. The particle is projected horizontally from \(A\) with speed \(\frac { 1 } { 2 } \sqrt { } ( g a )\). The particle leaves the surface of the sphere at the point \(C\), where \(\angle A O C = \theta\), and strikes the plane at the point \(P\), as shown in Figure 5.

1. Show that \(\cos \theta = \frac { 3 } { 4 }\).
2. Find the angle that the velocity of the particle makes with the horizontal as it reaches \(P\).

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)

4. \begin{figure}[h] \end{figure} A particle of mass \(3 m\) 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 \(O\). The particle is held at the point \(A\), where \(O A\) is horizontal and \(O A = a\). The particle is projected vertically downwards from \(A\) with speed \(u\), as shown in Figure 2. The particle moves in complete vertical circles.

1. Show that \(u ^ { 2 } \geqslant 3 a g\). Given that the greatest tension in the string is three times the least tension in the string, (b) show that \(u ^ { 2 } = 6 a g\).
   

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7. \begin{figure}[h] \end{figure} A particle of mass \(m\) is attached to one end of a light rod of length \(l\). The other end of the rod is attached to a fixed point \(O\). The rod can turn freely in a vertical plane about a horizontal axis through \(O\). The particle is projected with speed \(u\) from a point \(A\), where \(O A\) makes an angle \(\alpha\) with the upward vertical through \(O\), as shown in Figure 4. The particle moves in complete vertical circles. Given that \(\cos \alpha = \frac { 4 } { 5 }\)

1. show that \(u > \sqrt { \frac { 2 g l } { 5 } }\) As the rod rotates, the least tension in the rod is \(T\) and the greatest tension is \(4 T\).
2. Show that \(u = \sqrt { \frac { 17 } { 5 } g l }\)
   \includegraphics[max width=\textwidth, alt={}]{ffe0bc72-3136-48d9-9d5b-4a364d134070-12_2639_1830_121_121}

1 A particle \(P\) of mass 0.6 kg is attached to a fixed point \(O\) by a light inextensible string of length 0.4 m . While hanging at a distance 0.4 m vertically below \(O , P\) is projected horizontally with speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and moves in a complete vertical circle. Calculate the tension in the string when \(P\) is vertically above \(O\).

1 A particle \(P\) of mass 0.6 kg is attached to a fixed point \(O\) by a light inextensible string of length 0.4 m . While hanging at a distance 0.4 m vertically below \(O , P\) is projected horizontally with speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and moves in a complete vertical circle. Calculate the tension in the string when \(P\) is vertically above \(O\).

4 One end of a light inextensible string of length \(a\) is attached to a fixed point \(O\). A particle of mass \(m\) is attached to the other end of the string and is held with the string taut at the point \(A\). At \(A\) the string makes an angle \(\theta\) with the upward vertical through \(O\). The particle is projected perpendicular to the string in a downward direction from \(A\) with a speed \(u\). It moves along a circular path in the vertical plane. When the string makes an angle \(\alpha\) with the downward vertical through \(O\), the speed of the particle is \(2 u\) and the magnitude of the tension in the string is 10 times its magnitude at \(A\). It is given that \(\mathrm { u } = \sqrt { \frac { 2 } { 3 } \mathrm { ga } }\).

1. Find, in terms of \(m\) and \(g\), the magnitude of the tension in the string at \(A\).
2. Find the value of \(\cos \alpha\).

2 One end of a light inextensible string of length 0.5 m is attached to a fixed point \(A\). A particle \(P\) of mass 0.2 kg is attached to the other end of the string. \(P\) moves with constant speed in a horizontal circle with centre \(O\) which is 0.4 m vertically below \(A\).

1. Show that the tension in the string is 2.5 N .
2. Find the speed of \(P\).

1
\includegraphics[max width=\textwidth, alt={}, center]{9d377c95-09b8-4893-b29f-8517a5016e8b-2_381_1079_255_534} A particle \(P\) of mass 0.4 kg is attached to a fixed point \(A\) by a light inextensible string. The string is inclined at \(60 ^ { \circ }\) to the vertical. \(P\) moves with constant speed in a horizontal circle of radius 0.2 m . The centre of the circle is vertically below \(A\) (see diagram).

1. Show that the tension in the string is 8 N .
2. Calculate the speed of the particle.

5. A cone of semi-vertical angle \(60 ^ { \circ }\) is fixed with its axis vertical and vertex upwards. A particle 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 vertically above the vertex of the cone. The particle moves in a horizontal circle on the smooth outer surface of the cone with constant angular speed \(\omega\), with the string making a constant angle \(60 ^ { \circ }\) with the horizontal, as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{5f9a87c6-2255-4178-ab04-441bb0cc4ce0-10_538_648_456_664}

1. Find the tension in the string, in terms of \(m , l , \omega\) and \(g\). The particle remains on the surface of the cone.
2. Show that the time for the particle to make one complete revolution is greater than $$2 \pi \sqrt { \frac { l \sqrt { 3 } } { 2 g } } .$$ [Question 5 Continued]
   [0pt] [Question 5 Continued]

7 A light inextensible string, of length \(a\), has one end attached to a fixed point \(O\). A particle, of mass \(m\), is attached to the other end. The particle is moving in a vertical circle, centre \(O\). When the particle is at \(B\), vertically above \(O\), the string is taut and the particle is moving with speed \(3 \sqrt { a g }\).
\includegraphics[max width=\textwidth, alt={}, center]{1bc18163-b20e-4dc6-bd35-496efec8dc73-5_422_399_497_778}

1. Find, in terms of \(g\) and \(a\), the speed of the particle at the lowest point, \(A\), of its path.
2. Find, in terms of \(g\) and \(m\), the tension in the string when the particle is at \(A\).

1
\includegraphics[max width=\textwidth, alt={}, center]{5960a9cf-2c51-4c07-9973-c29604762df7-2_540_269_395_897} 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.

5 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 \(O\). The string is held taut with \(O P\) horizontal. The particle \(P\) is projected vertically downwards with speed \(\sqrt { \frac { 1 } { 3 } \mathrm { ag } }\) and starts to move in a vertical circle. \(P\) passes through the lowest point of the circle and reaches the point \(Q\) where \(O Q\) makes an angle \(\theta\) with the downward vertical. At \(Q\) the speed of \(P\) is \(\sqrt { \mathrm { kag } }\) and the tension in the string is \(\frac { 11 } { 6 } \mathrm { mg }\).

1. Find the value of \(k\) and the value of \(\cos \theta\).
   At \(Q\) the particle \(P\) becomes detached from the string.
2. In the subsequent motion, find the greatest height reached by \(P\) above the level of the lowest point of the circle.

6. A particle \(P\) is free to move on the smooth inner surface of a fixed thin hollow sphere of internal radius \(a\) and centre \(O\). The particle passes through the lowest point of the spherical surface with speed \(U\). The particle loses contact with the surface when \(O P\) is inclined at an angle \(\alpha\) to the upward vertical.

1. Show that \(\quad U ^ { 2 } = a g ( 2 + 3 \cos \alpha )\). The particle has speed \(W\) as it passes through the level of \(O\). Given that \(\cos \alpha = \frac { 1 } { \sqrt { } 3 }\), (b) show that \(\quad W ^ { 2 } = a g \sqrt { } 3\).

6. \begin{figure}[h] \end{figure} Figure 5 shows a hollow sphere, with centre \(O\) and internal radius \(a\), which is fixed to a horizontal surface. A particle \(P\) of mass \(m\) is projected horizontally with speed \(\sqrt { \frac { 7 a g } { 2 } }\) from the lowest point \(A\) of the inner surface of the sphere. The particle moves in a vertical circle with centre \(O\) on the smooth inner surface of the sphere. The particle passes through the point \(B\), on the inner surface of the sphere, where \(O B\) is horizontal.

1. Find, in terms of \(m\) and \(g\), the normal reaction exerted on \(P\) by the surface of the sphere when \(P\) is at \(B\). The particle leaves the inner surface of the sphere at the point \(C\), where \(O C\) makes an angle \(\theta , \theta > 0\), with the upward vertical.
2. Show that, after leaving the surface of the sphere at \(C\), the particle is next in contact with the surface at \(A\).

   END

A particle \(P\) of mass \(m\) is free to move on the smooth inner surface of a fixed hollow sphere of radius \(a\). The centre of the sphere is \(O\). The points \(A\) and \(A ^ { \prime }\) are on the inner surface of the sphere, on opposite sides of the vertical through \(O\); the radius \(O A\) makes an angle \(\alpha\) with the downward vertical and the radius \(O A ^ { \prime }\) makes an angle \(\beta\) with the upward vertical. The point \(B\) is on the inner surface of the sphere, vertically below \(O\). The point \(B ^ { \prime }\) is on the inner surface of the sphere and such that \(O B ^ { \prime }\) makes an angle \(2 \beta\) with the upward vertical through \(O\) (see diagram). It is given that \(\cos \alpha = \frac { 1 } { 16 }\).

1. \(P\) is projected from \(A\) with speed \(u\) along the surface of the sphere downwards towards \(B\). Subsequently it loses contact with the sphere at \(A ^ { \prime }\). Show that \(u ^ { 2 } = \frac { 1 } { 8 } a g ( 1 + 24 \cos \beta )\).
2. \(P\) is now projected from \(B\) with speed \(u\) along the surface of the sphere towards \(B ^ { \prime }\). Subsequently it loses contact with the sphere at \(B ^ { \prime }\). Find \(\cos \beta\).
3. In part (i), the reaction of the sphere on \(P\) when it is initially projected at \(A\) is \(R\). Find \(R\) in terms of \(m\) and \(g\).

7

1. State two reasons why Ben's diagram is not a good representation of the situation. Reason 1 \(\_\_\_\_\)
   Reason 2 \(\_\_\_\_\)
   7
2. Using your answer to part (a), sketch an improved diagram.
   \includegraphics[max width=\textwidth, alt={}, center]{4975a2a9-1e45-44b4-b525-fc902627d03e-11_791_132_356_954} Question 7 continues on the next page 7
3. \(\quad\) Find \(\alpha\), giving your answer to the nearest degree.
   \includegraphics[max width=\textwidth, alt={}, center]{4975a2a9-1e45-44b4-b525-fc902627d03e-13_2488_1716_219_153}

6. \begin{figure}[h] \end{figure} A particle \(P\) 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 \(O\). The particle is held at the point \(A\), where \(O A = a\) and \(O A\) is horizontal. The point \(B\) is vertically above \(O\) and the point \(C\) is vertically below \(O\), with \(O B = O C = a\), as shown in Figure 5. The particle is projected vertically upwards with speed \(3 \sqrt { } ( a g )\).

1. Show that \(P\) will pass through \(B\).
2. Find the speed of \(P\) as it reaches \(C\). As \(P\) passes through \(C\) it receives an impulse. Immediately after this, the speed of \(P\) is \(\frac { 5 } { 12 } \sqrt { } ( 11 a g )\) and the direction of motion of \(P\) is unchanged.
3. Find the angle between the string and the downward vertical when \(P\) comes to 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}

4 One end of a light inextensible string of length 0.5 m is attached to a fixed point \(A\). The other end of the string is attached to a particle \(P\) of weight 6 N . Another light inextensible string of length 0.5 m connects \(P\) to a fixed point \(B\) which is 0.8 m vertically below \(A\). The particle \(P\) moves with constant speed in a horizontal circle with centre at the mid-point of \(A B\). Both strings are taut.

1. Calculate the speed of \(P\) when the tension in the string \(B P\) is 2 N .
2. Show that the angular speed of \(P\) must exceed \(5 \mathrm { rad } \mathrm { s } ^ { - 1 }\).

2 A particle \(P\) is projected with speed \(26 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(30 ^ { \circ }\) above the horizontal from a point \(O\) on a horizontal plane.

1. For the instant when the vertical component of the velocity of \(P\) is \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) downwards, find the direction of motion of \(P\) and the height of \(P\) above the plane.
2. \(P\) strikes the plane at the point \(A\). Calculate the time taken by \(P\) to travel from \(O\) to \(A\) and the distance \(O A\).
   \includegraphics[max width=\textwidth, alt={}, center]{022776e2-80bb-4e73-a309-8cd972ae7377-2_679_455_1544_845} Particles \(P\) and \(Q\) have masses 0.8 kg and 0.4 kg respectively. \(P\) is attached to a fixed point \(A\) by a light inextensible string which is inclined at an angle \(\alpha ^ { \circ }\) to the vertical. \(Q\) is attached to a fixed point \(B\), which is vertically below \(A\), by a light inextensible string of length 0.3 m . The string \(B Q\) is horizontal. \(P\) and \(Q\) are joined to each other by a light inextensible string which is vertical. The particles rotate in horizontal circles of radius 0.3 m about the axis through \(A\) and \(B\) with constant angular speed \(5 \mathrm { rad } \mathrm { s } ^ { - 1 }\) (see diagram).
3. By considering the motion of \(Q\), find the tensions in the strings \(P Q\) and \(B Q\).
4. Find the tension in the string \(A P\) and the value of \(\alpha\).

7. A particle \(P\) of mass \(m \mathrm {~kg}\) is attached to points \(C\) and \(D\) on the same horizontal level by means of two light inextensible strings \(C P\) and \(D P\), both of length \(40 \mathrm {~cm} . P\) is projected with speed \(u \mathrm {~ms} ^ { - 1 }\) so as to move in a vertical circle in a plane perpendicular to \(C D\), so that angle \(P C D =\) angle \(P D C = \theta\) throughout the motion.
\includegraphics[max width=\textwidth, alt={}, center]{627b3411-07ba-4ee4-a672-93a64eeb90b3-2_335_405_1775_1572} If \(u\) is just large enough for the strings to remain taut as \(P\) describes this circular path,

1. show that \(u ^ { 2 } = 2 g \sin \theta\). The string \(D P\) breaks when \(P\) is at its lowest point. \(P\) then immediately starts to move in a horizontal circle on the end of the string \(C P\).
2. Prove that \(\tan \theta = \frac { 1 } { 5 } \sqrt { 5 }\).

2 A hollow hemispherical bowl of radius \(a\) has a smooth inner surface and is fixed with its axis vertical. A particle \(P\) of mass \(m\) moves in horizontal circles on the inner surface of the bowl, at a height \(x\) above the lowest point of the bowl. The speed of \(P\) is \(\sqrt { \frac { 8 } { 3 } } \mathrm { ga }\). Find \(x\) in terms of \(a\).

2. \begin{figure}[h] \end{figure} A hemispherical bowl of internal radius \(6 r\) is fixed with its circular rim horizontal. The centre of the circular rim is \(O\) and the point \(A\) on the surface of the bowl is vertically below \(O\). A particle \(P\) moves in a horizontal circle, with centre \(C\), on the smooth inner surface of the bowl. The particle moves with constant angular speed \(\sqrt { \frac { g } { 4 r } }\). The point \(C\) lies on \(O A\), as shown in Figure 1. Find, in terms of \(r\), the distance \(O C\)

2
\includegraphics[max width=\textwidth, alt={}, center]{7fcedc6d-8dc1-4159-8a72-be0f6a3f659b-2_698_737_484_703} A particle \(P\) travels on a smooth surface whose vertical cross-section is in the form of two arcs of circles. The first arc \(A B\) is a quarter of a circle of radius \(\frac { 1 } { 8 } a\) and centre \(O\). The second arc \(B C\) is a quarter of a circle of radius \(a\) and centre \(Q\). The two arcs are smoothly joined at \(B\). The point \(Q\) is vertically below \(O\) and the two arcs are in the same vertical plane. The particle \(P\) is projected vertically downwards from \(A\) with speed \(u\). When \(P\) is on the \(\operatorname { arc } B C\), angle \(B Q P\) is \(\theta\) (see diagram). Given that \(P\) loses contact with the surface when \(\cos \theta = \frac { 5 } { 6 }\), find \(u\) in terms of \(a\) and \(g\).

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. \begin{figure}[h] \end{figure} A particle \(P\) of mass \(m\) is attached to one end of a light rod of length \(l\). The other end of the rod is attached to a fixed point \(O\). The rod can rotate freely in a vertical plane about \(O\). The particle is projected with speed \(u\) from a point \(A\). The line \(O A\) makes an angle \(\alpha\) with the upward vertical through \(O\), where \(\alpha < \frac { \pi } { 2 }\) When \(O P\) makes an angle \(\theta\) with the upward vertical through \(O\), the speed of \(P\) is \(v\), as shown in Figure 4.

1. Show that \(v ^ { 2 } = u ^ { 2 } - 2 g l ( \cos \theta - \cos \alpha )\) Given that \(\cos \alpha = \frac { 2 } { 5 }\) and that \(u = \sqrt { 3 g l }\)
2. show that \(P\) moves in a complete vertical circle. As the rod rotates, the least tension in the rod is \(T\) and the greatest tension is \(k T\)
3. Find the exact value of \(k\)

4. \begin{figure}[h] \end{figure} A circus performer has mass \(m\). She is attached to one end of a cable of length \(l\). The other end of the cable is attached to a fixed point \(O\) Initially she is held at rest at point \(A\) with the cable taut and at an angle of \(30 ^ { \circ }\) below the horizontal, as shown in Figure 3. The circus performer is released from \(A\) and she moves on a vertical circular path with centre \(O\) The circus performer is modelled as a particle and the cable is modelled as light and inextensible.

1. Find, in terms of \(m\) and \(g\), the tension in the cable at the instant immediately after the circus performer is released.
2. Show that, during the motion following her release, the greatest tension in the cable is 4 times the least tension in the cable.

1. A particle, \(P\), of mass \(m\) is attached to one end of a light rod of length \(L\). The other end of the rod is attached to a fixed point \(O\) so that the rod is free to rotate in a vertical plane about \(O\). The particle is held with the rod horizontal and is then projected vertically downwards with speed \(u\). The particle first comes to instantaneous rest at the point \(A\).
   1. Explain why the acceleration of \(P\) at \(A\) is perpendicular to \(O A\).
   At the instant when \(P\) is at the point \(A\) the acceleration of \(P\) is in a direction making an angle \(\theta\) with the horizontal. Given that \(u ^ { 2 } = \frac { 2 g L } { 3 }\),
2. find
   1. the magnitude of the acceleration of \(P\) at the point \(A\),
   2. the size of \(\theta\).
3. Find, in terms of \(m\) and \(g\), the magnitude of the tension in the rod at the instant when \(P\) is at its lowest point.

3 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 \(O\). The path of the particle is a complete vertical circle with centre \(O\). When \(P\) is at its lowest point, its speed is \(u\). When \(P\) is at the point \(A\), the tension in the string is \(T\) and the string makes an angle \(\theta\) with the downward vertical, where \(\cos \theta = \frac { 3 } { 5 }\). When \(P\) is at the point \(B\), above the level of \(O\), the tension in the string is \(\frac { 1 } { 8 } T\) and angle \(B O A = 90 ^ { \circ }\). Find \(u\) in terms of \(a\) and \(g\).

1. By considering the total energy of the system, obtain an expression for \(v ^ { 2 }\) in terms of \(\theta\).
2. Show that the magnitude of the force exerted on \(P\) by the cylinder is \(( 7.12 \sin \theta - 4.64 \theta ) \mathrm { N }\).
3. Given that \(P\) leaves the surface of the cylinder when \(\theta = \alpha\), show that \(1.53 < \alpha < 1.54\).

7 A hollow cylinder of radius \(a\) is fixed with its axis horizontal. A particle \(P\), of mass \(m\), moves in part of a vertical circle of radius \(a\) and centre \(O\) on the smooth inner surface of the cylinder. The speed of \(P\) when it is at the lowest point \(A\) of its motion is \(\sqrt { \frac { 7 } { 2 } \mathrm { ga } }\). The particle \(P\) loses contact with the surface of the cylinder when \(O P\) makes an angle \(\theta\) with the upward vertical through \(O\).

1. Show that \(\theta = 60 ^ { \circ }\).
2. Show that in its subsequent motion \(P\) strikes the cylinder at the point \(A\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

3
\includegraphics[max width=\textwidth, alt={}, center]{36259e2a-aa9b-4655-b0c2-891f96c3f5a4-3_637_572_264_788} One end of a light inextensible string is attached to a point \(C\). The other end is attached to a point \(D\), which is 1.1 m vertically below \(C\). A small smooth ring \(R\), of mass 0.2 kg , is threaded on the string and moves with constant speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a horizontal circle, with centre at \(O\) and radius 1.2 m , where \(O\) is 0.5 m vertically below \(D\) (see diagram).

1. Show that the tension in the string is 1.69 N , correct to 3 significant figures.
2. Find the value of \(v\).

2 A particle \(P\) of mass \(m\) is attached to one end of a light elastic string of natural length \(a\) and modulus of elasticity \(2 m g\). A particle \(Q\) of mass \(k m\) is attached to the other end of the string. Particle \(P\) lies on a smooth horizontal table. The string has part of its length in contact with the table and then passes through a small smooth hole \(H\) in the table. Particle \(P\) moves in a horizontal circle on the surface of the table with constant speed \(\sqrt { \frac { 1 } { 2 } g a }\). Particle \(Q\) hangs in equilibrium vertically below the hole with \(H Q = \frac { 1 } { 4 } a\).

1. Find, in terms of \(a\), the extension in the string.
2. Find the value of \(k\).

1. 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 \(O\). The particle is released from rest with the string taut and \(O P\) horizontal.
   1. Find the tension in the string when \(O P\) makes an angle of \(60 ^ { \circ }\) with the downward vertical.
   A particle \(Q\) of mass \(3 m\) is at rest at a distance \(a\) vertically below \(O\). When \(P\) strikes \(Q\) the particles join together and the combined particle of mass \(4 m\) starts to move in a vertical circle with initial speed \(u\).
2. Show that \(u = \sqrt { } \left( \frac { g a } { 8 } \right)\). The combined particle comes to instantaneous rest at \(A\).
3. Find
   1. the angle that the string makes with the downward vertical when the combined particle is at \(A\),
   2. the tension in the string when the combined particle is at \(A\).
      \section*{LU
      \(\_\_\_\_\)}

4
\includegraphics[max width=\textwidth, alt={}, center]{ab5f2781-e5ce-4fce-bc95-9d7f55ea66d9-2_515_583_1388_781} A smooth wire is in the form of an \(\operatorname { arc } A B\) of a circle, of radius \(a\), that subtends an obtuse angle \(\pi - \theta\) at the centre \(O\) of the circle. It is given that \(\sin \theta = \frac { 1 } { 4 }\). The wire is fixed in a vertical plane, with \(A O\) horizontal and \(B\) below the level of \(O\) (see diagram). A small bead of mass \(m\) is threaded on the wire and projected vertically downwards from \(A\) with speed \(\sqrt { } \left( \frac { 3 } { 10 } g a \right)\).

1. Find the reaction between the bead and the wire when the bead is vertically below \(O\).
2. Find the speed of the bead as it leaves the wire at \(B\).
3. Show that the greatest height reached by the bead is \(\frac { 1 } { 8 } a\) above the level of \(O\).

5
\includegraphics[max width=\textwidth, alt={}, center]{fe5c198d-5d05-4241-98f5-894ba92f7afe-3_593_828_1530_660} A horizontal disc of radius 0.5 m is rotating with constant angular speed \(\omega \mathrm { rad } \mathrm { s } ^ { - 1 }\) about a fixed vertical axis through its centre \(O\). One end of a light inextensible string of length 0.8 m is attached to a point \(A\) of the circumference of the disc. A particle \(P\) of mass 0.4 kg is attached to the other end of the string. The string is taut and the system rotates so that the string is always in the same vertical plane as the radius \(O A\) of the disc. The string makes a constant angle \(\theta\) with the vertical (see diagram). The speed of \(P\) is 1.6 times the speed of \(A\).

1. Show that \(\sin \theta = \frac { 3 } { 8 }\).
2. Find the tension in the string.
3. Find the value of \(\omega\).

1 The end \(A\) of a \(\operatorname { rod } A B\) of length 1.2 m is freely pivoted at a fixed point. The rod rotates about \(A\) in a vertical plane. Calculate the angular speed of the rod at an instant when \(B\) has speed \(0.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).