6.05f Vertical circle: motion including free fall

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 \(\cos \alpha\).

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 particle \(P\) is held at the point \(A\), where \(O A\) makes an angle \(\theta\) with the downward vertical through \(O\), and with the string taut. The particle \(P\) is projected perpendicular to \(O A\) in an upwards direction with speed \(u\). It then starts to move along a circular path in a vertical plane. The string goes slack when \(P\) is at \(B\), where angle \(A O B\) is \(90 ^ { \circ }\) and the speed of \(P\) is \(\sqrt { \frac { 4 } { 5 } \mathrm { ag } }\).

1. Find the value of \(\sin \theta\).
2. Find, in terms of \(m\) and \(g\), the tension in the string when \(P\) is at \(A\). \includegraphics[max width=\textwidth, alt={}, center]{454be64a-204f-4fa4-a5fc-72fd88e1289f-06_846_767_258_689} An object is formed from a solid hemisphere, of radius \(2 a\), and a solid cylinder, of radius \(a\) and height \(d\). The hemisphere and the cylinder are made of the same material. The cylinder is attached to the plane face of the hemisphere. The line \(O C\) forms a diameter of the base of the cylinder, where \(C\) is the centre of the plane face of the hemisphere and \(O\) is common to both circumferences (see diagram). Relative to axes through \(O\), parallel and perpendicular to \(O C\) as shown, the centre of mass of the object is ( \(\mathrm { x } , \mathrm { y }\) ).

3. A light rod \(A B\) of length \(2 a\) has a particle \(P\) of mass \(m\) attached to \(B\). The rod is rotating in a vertical plane about a fixed smooth horizontal axis through \(A\). Given that the greatest tension in the rod is \(\frac { 9 m g } { 8 }\), find, to the nearest degree, the angle between the rod and the downward vertical when the speed of \(P\) is \(\sqrt { \left( \frac { a g } { 20 } \right) }\).

6. A smooth sphere, with centre \(O\) and radius \(a\), is fixed with its lowest point \(A\) on a horizontal floor. A particle \(P\) is placed on the surface of the sphere at the point \(B\), where \(B\) is vertically above \(A\). The particle is projected horizontally from \(B\) with speed \(\sqrt { \frac { a g } { 5 } }\) and moves along the surface of the sphere. When \(O P\) makes an angle \(\theta\) with the upward vertical, and \(P\) is still in contact with the sphere, the speed of \(P\) is \(v\).

1. Show that \(v ^ { 2 } = \frac { a g } { 5 } ( 11 - 10 \cos \theta )\). The particle leaves the surface of the sphere at the point \(C\).
   Find
2. the speed of \(P\) at \(C\) in terms of \(a\) and \(g\),
3. the size of the angle between the floor and the direction of motion of \(P\) at the instant immediately before \(P\) hits the floor.

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.

6. \begin{figure}[h] \end{figure} 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 held at the point \(A\), where \(O A = a\) and \(O A\) is horizontal, as shown in Figure 4. The particle is projected vertically downwards with speed \(\sqrt { \frac { 9 a g } { 5 } }\) When the string makes an angle \(\theta\) with the downward vertical through \(O\) and the string is still taut, the tension in the string is \(S\).

1. Show that \(S = \frac { 3 } { 5 } m g ( 5 \cos \theta + 3 )\) At the instant when the string becomes slack, the speed of \(P\) is \(v\)
2. Show that \(v = \sqrt { \frac { 3 a g } { 5 } }\)
3. Find the maximum height of \(P\) above the horizontal level of \(O\)

6. \begin{figure}[h] \end{figure} 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 \(P\) is held at rest with the string taut and horizontal and is then projected vertically downwards with speed \(u\), as shown in Figure 5. Air resistance is modelled as being negligible.
At the instant when the string has turned through an angle \(\theta\) and the string is taut, the tension in the string is \(T\).

1. Show that \(T = \frac { m u ^ { 2 } } { a } + 3 m g \sin \theta\) Given that \(u = 2 \sqrt { \frac { 3 a g } { 5 } }\)
2. find, in terms of \(a\) and \(g\), the speed of \(P\) at the instant when the string goes slack.
3. Hence find, in terms of \(a\), the maximum height of \(P\) above \(O\) in the subsequent motion.

6. \begin{figure}[h] \end{figure} A fixed solid sphere has centre \(O\) and radius \(r\).
A particle \(P\) of mass \(m\) is held at rest on the smooth surface of the sphere at \(A\), the highest point of the sphere.
The particle \(P\) is then projected horizontally from \(A\) with speed \(u\) and moves on the surface of the sphere.
At the instant when \(P\) reaches the point \(B\) on the sphere, where angle \(A O B = \theta , P\) is moving with speed \(v\), as shown in Figure 4. At this instant, \(P\) loses contact with the surface of the sphere.

1. Show that $$\cos \theta = \frac { 2 g r + u ^ { 2 } } { 3 g r }$$ In the subsequent motion, the particle \(P\) crosses the horizontal through \(O\) at the point \(C\), also shown in Figure 4. At the instant \(P\) passes through \(C , P\) is moving at an angle \(\alpha\) to the horizontal.
   Given that \(u ^ { 2 } = \frac { 2 g r } { 5 }\)
2. find the exact value of \(\tan \alpha\).

6. \begin{figure}[h] \end{figure} Figure 3 represents the path of a skier of mass 70 kg moving on a ski-slope \(A B C D\). The path lies in a vertical plane. From \(A\) to \(B\), the path is modelled as a straight line inclined at \(60 ^ { \circ }\) to the horizontal. From \(B\) to \(D\), the path is modelled as an arc of a vertical circle of radius 50 m . The lowest point of the \(\operatorname { arc } B D\) is \(C\). At \(B\), the skier is moving downwards with speed \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). At \(D\), the path is inclined at \(30 ^ { \circ }\) to the horizontal and the skier is moving upwards. By modelling the slope as smooth and the skier as a particle, find

1. the speed of the skier at \(C\),
2. the normal reaction of the slope on the skier at \(C\),
3. the speed of the skier at \(D\),
4. the change in the normal reaction of the slope on the skier as she passes \(B\). The model is refined to allow for the influence of friction on the motion of the skier.
5. State briefly, with a reason, how the answer to part (b) would be affected by using such a model. (No further calculations are expected.)

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)

6. 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. The particle is hanging freely at rest, with the string vertical, when it is projected horizontally with speed \(U\). The particle moves in a complete vertical circle.

1. Show that \(U \geqslant \sqrt { 5 a g }\) As \(P\) moves in the circle the least tension in the string is \(T\) and the greatest tension is \(k T\). Given that \(U = 3 \sqrt { a g }\)
2. find the value of \(k\).

6. One end of a light inextensible string of length \(l\) is attached to a particle \(P\) of mass \(2 m\). The other end of the string is attached to a fixed point \(A\). The particle is hanging freely at rest with the string vertical. The particle is then projected horizontally with speed \(\sqrt { \frac { 7 g l } { 2 } }\) (a) Find the speed of \(P\) at the instant when the string is horizontal.
(4) When the string is horizontal and \(P\) is moving upwards, the string comes into contact with a small smooth peg which is fixed at the point \(B\), where \(A B\) is horizontal and \(A B < l\). The particle then describes a complete semicircle with centre \(B\).
(b) Show that \(A B \geqslant \frac { 1 } { 2 } l\)

5. \begin{figure}[h] \end{figure} A hollow cylinder is fixed with its axis horizontal. A particle \(P\) moves in a vertical circle, with centre \(O\) and radius \(a\), on the smooth inner surface of the cylinder. The particle moves in a vertical plane which is perpendicular to the axis of the cylinder. The particle is projected vertically downwards with speed \(\sqrt { 7 a g }\) from the point \(A\), where \(O A\) is horizontal and \(O A = a\). When angle \(A O P = \theta\), the speed of \(P\) is \(v\), as shown in Figure 4.

1. Show that \(v ^ { 2 } = a g ( 7 + 2 \sin \theta )\)
2. Verify that \(P\) will move in a complete circle.
3. Find the maximum value of \(v\).

4 A particle \(P\) of mass \(m\) is suspended from a fixed point \(O\) by a light inextensible string of length \(a\). When hanging at rest under gravity, \(P\) is given a horizontal velocity of magnitude \(\sqrt { } ( 3 a g )\) and subsequently moves freely in a vertical circle. Show that the tension \(T\) in the string when \(O P\) makes an angle \(\theta\) with the downward vertical is given by $$T = m g ( 1 + 3 \cos \theta )$$ When the string is horizontal, it comes into contact with a small smooth peg \(Q\) which is at the same horizontal level as \(O\) and at a distance \(x\) from \(O\), where \(x < a\). Given that \(P\) completes a vertical circle about \(Q\), find the least possible value of \(x\).

4 A smooth sphere, with centre \(O\) and radius \(a\), has its lowest point fixed on a horizontal plane. A particle \(P\) of mass \(m\) is projected horizontally with speed \(u\) from the highest point on the outer surface of the sphere. In the subsequent motion, \(O P\) makes an angle \(\theta\) with the upward vertical through \(O\). Show that, while \(P\) remains in contact with the sphere, the magnitude of the reaction of the sphere on \(P\) is \(m g ( 3 \cos \theta - 2 ) - \frac { m u ^ { 2 } } { a }\). The particle loses contact with the surface of the sphere when \(\theta = \alpha\). Given that \(u = \frac { 1 } { 2 } \sqrt { } ( g a )\), find

1. \(\cos \alpha\),
2. the vertical component of the velocity of \(P\) as it strikes the horizontal plane.

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 \(O\). When \(P\) is hanging vertically below \(O\), it is given a horizontal speed \(u\). In the subsequent motion, \(P\) moves in a complete circle. When \(O P\) makes an angle \(\theta\) with the downward vertical, the tension in the string is \(T\). Show that $$T = \frac { m u ^ { 2 } } { a } + m g ( 3 \cos \theta - 2 )$$ Given that the ratio of the maximum value of \(T\) to the minimum value of \(T\) is \(3 : 1\), find \(u\) in terms of \(a\) and \(g\). Assuming this value of \(u\), find the value of \(\cos \theta\) when the tension is half of its maximum value.

A light elastic string has modulus of elasticity \(\frac { 3 } { 2 } m g\) and natural length \(a\). A particle of mass \(m\) is attached to one end of the string. The other end of the string is attached to a fixed point \(A\). The particle is released from rest at \(A\). Show that when the particle has fallen a distance \(k a\) from \(A\), where \(k > 1\), its kinetic energy is $$\frac { 1 } { 4 } m g a \left( 10 k - 3 - 3 k ^ { 2 } \right) .$$ Show that the particle first comes to instantaneous rest at the point \(B\) which is at a distance \(3 a\) vertically below \(A\). Show that the time taken by the particle to travel from \(A\) to \(B\) is $$\sqrt { } \left( \frac { 2 a } { g } \right) + \frac { 2 \pi } { 3 } \sqrt { } \left( \frac { 2 a } { 3 g } \right)$$

5 \includegraphics[max width=\textwidth, alt={}, center]{7fcedc6d-8dc1-4159-8a72-be0f6a3f659b-3_355_693_260_726} \(A B C D\) is a uniform rectangular lamina of mass \(m\) in which \(A B = 4 a\) and \(B C = 2 a\). The lines \(A C\) and \(B D\) intersect at \(O\). The mid-points of \(O A , O B , O C , O D\) are \(E , F , G , H\) respectively. The rectangle \(E F G H\), in which \(E F = 2 a\) and \(F G = a\), is removed from \(A B C D\) (see diagram). The resulting lamina is free to rotate in a vertical plane about a fixed smooth horizontal axis through \(A\) and perpendicular to the plane of \(A B C D\). Show that the moment of inertia of this lamina about the axis is \(\frac { 85 } { 16 } m a ^ { 2 }\). The lamina hangs in equilibrium under gravity with \(C\) vertically below \(A\). The point \(C\) is now given a speed \(u\). Given that the lamina performs complete revolutions, show that $$u ^ { 2 } > \frac { 192 \sqrt { } 5 } { 17 } a g .$$

2 The point \(O\) is on the fixed line \(l\). The point \(A\) on \(l\) is such that \(O A = 3 \mathrm {~m}\). A particle \(P\) oscillates on \(l\) in simple harmonic motion with centre \(O\) and period \(\pi\) seconds. When \(P\) is at \(A\) its speed is \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the speed of \(P\) when it is at the point \(B\) on \(l\), where \(O B = 6 \mathrm {~m}\) and \(B\) is on the same side of \(O\) as \(A\). Find, correct to 2 decimal places, the time, in seconds, taken for \(P\) to travel directly from \(A\) to \(B\).

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 \(u\).

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 \(P\).

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 6 kg is attached to one end of a light inextensible rod of length 2.4 m . The other end of the rod is smoothly hinged at a fixed point \(O\) and the rod is free to rotate in any direction. Initially, \(P\) is at rest, vertically below \(O\), when it is projected horizontally with a speed of \(12 \mathrm {~ms} ^ { - 1 }\). It subsequently describes complete vertical circles with \(O\) as the centre. \includegraphics[max width=\textwidth, alt={}, center]{05b479a4-4087-4332-924b-43b1aedbb4f2-3_611_517_536_246} The angle that the rod makes with the downward vertical through \(O\) at each instant is denoted by \(\theta\) and \(A\) is the point which \(P\) passes through where \(\theta = 40 ^ { \circ }\) (see diagram).

1. Find the tangential acceleration of \(P\) at \(A\), stating its direction.
2. Determine the radial acceleration of \(P\) at \(A\), stating its direction.
3. Find the magnitude of the force in the rod when \(P\) is at \(A\), stating whether the rod is in tension or compression. The motion is now stopped when \(P\) is at \(A\), and \(P\) is then projected in such a way that it now describes horizontal circles at a constant speed with \(\theta = 40 ^ { \circ }\) (see diagram). \includegraphics[max width=\textwidth, alt={}, center]{05b479a4-4087-4332-924b-43b1aedbb4f2-3_403_524_1877_242}
4. Find the speed of \(P\).
5. Explain why, wherever \(P\) 's motion is initiated from and whatever its initial velocity, it is not possible for \(P\) to describe horizontal circles at constant speed with \(\theta = 90 ^ { \circ }\).