Vertical circle: complete revolution conditions

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}

6. \begin{figure}[h] \end{figure} A particle of mass \(m\) is attached to one end of a light inextensible string of length \(2 a\). The other end of the string is attached to a fixed point \(O\). The particle is initially held at the point \(A\) with the string taut and \(O A\) making an angle of \(60 ^ { \circ }\) with the downward vertical. The particle is then projected upwards with a speed of \(3 \sqrt { a g }\), perpendicular to \(O A\), in the vertical plane containing \(O A\), as shown in Figure 7. In an initial model of the motion of the particle, it is assumed that the string does not break. Using this model,

1. show that the particle performs complete vertical circles. In a refined model it is assumed that the string will break if the tension in it exceeds 7 mg . Using this refined model,
2. show that the particle still performs complete vertical circles.
   \includegraphics[max width=\textwidth, alt={}, center]{8a687d17-ec7e-463f-84dd-605f5c230db1-20_2249_50_314_1982}

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 \(u ^ { 2 } = 6 a g\).

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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5. \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 fixed at the point \(O\). The particle is initially held with \(O P\) horizontal and the string taut. It is then projected vertically upwards with speed \(u\), where \(u ^ { 2 } = 5 a g\). When \(O P\) has turned through an angle \(\theta\) the speed of \(P\) is \(v\) and the tension in the string is \(T\), as shown in Figure 5.

1. Find, in terms of \(a , g\) and \(\theta\), an expression for \(v ^ { 2 }\).
2. Find, in terms of \(m , g\) and \(\theta\), an expression for \(T\).
3. Prove that \(P\) moves in a complete circle.
4. Find the maximum speed of \(P\).

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 \(k\).

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 \(v\).

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. The particle is projected vertically upwards with speed \(u\), as shown in Figure 2. When the string makes an angle \(\theta\) with the horizontal through \(O\) and the string is still taut, the tension in the string is \(T\).

1. Show that \(T = \frac { m } { a } \left( u ^ { 2 } - 3 a g \sin \theta \right)\) The particle moves in complete circles.
2. Find, in terms of \(a\) and \(g\), the minimum value of \(u\). Given that the least tension in the string is \(S\) and the greatest tension in the string is \(4 S\),
3. find, in terms of \(a\) and \(g\), an expression for \(u\).

5. \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 fixed at the point \(O\). The particle is initially held with \(O P\) horizontal and the string taut. It is then projected vertically upwards with speed \(u\), where \(u ^ { 2 } = 5 a g\). When \(O P\) has turned through an angle \(\theta\) the speed of \(P\) is \(v\) and the tension in the string is \(T\), as shown in Figure 5.

1. Find, in terms of \(a , g\) and \(\theta\), an expression for \(v ^ { 2 }\).
2. Find, in terms of \(m , g\) and \(\theta\), an expression for \(T\).
3. Prove that \(P\) moves in a complete circle.
4. Find the maximum speed of \(P\).

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

8 A smooth wire is fixed in a vertical plane so that it forms a circle of radius \(a\) metres and centre \(O\). A bead, \(B\), of mass 0.3 kg , is threaded on the wire and is set in motion with a speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at the lowest point of its circular path, as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{31ba38f7-38a8-4e4e-96a3-19e819fabfb0-6_364_378_466_845}

1. Show that, if the bead is going to make complete revolutions around the wire, $$u > 2 \sqrt { a g }$$
2. At time \(t\) seconds, the angle between \(O B\) and the horizontal is \(\theta\), as shown in the diagram.
   \includegraphics[max width=\textwidth, alt={}, center]{31ba38f7-38a8-4e4e-96a3-19e819fabfb0-6_330_328_1231_858} It is given that \(u = \sqrt { \frac { 9 } { 2 } a g }\).
   1. Find the reaction of the bead on the wire, giving your answer in terms of \(g\) and \(\theta\).
   2. Find \(\theta\) when this reaction is zero.

5 A particle of mass \(m\) is suspended from a fixed point \(O\) by a light inextensible string of length \(l\). The particle hangs in equilibrium at the point \(R\) vertically below \(O\). The particle is set into motion with a horizontal velocity \(u\) so that it moves in a complete vertical circle with centre \(O\). The point \(T\) on the circle is such that angle \(R O T\) is \(30 ^ { \circ }\), as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{7c2c50e0-4976-4301-9898-61b2760a2aee-12_766_736_644_651}

1. Find, in terms of \(g , l\) and \(u\), the speed of the particle at the point \(T\).
2. Find, in terms of \(g , l , m\) and \(u\), the tension in the string when the particle is at the point \(T\).
3. Find, in terms of \(g , l , m\) and \(u\), the tension in the string when the particle returns to the point \(R\).
4. The particle makes complete revolutions. Find, in terms of \(g\) and \(l\), the minimum value of \(u\).
   [0pt] [4 marks]

7
\includegraphics[max width=\textwidth, alt={}, center]{2734e846-f640-4203-ac11-6b2180a21950-4_282_474_1809_794} One end of a light inextensible string of length 0.5 m is attached to a fixed point \(O\). A particle \(P\) of mass 0.2 kg is attached to the other end of the string. \(P\) is projected horizontally from the point 0.5 m below \(O\) with speed \(u \mathrm {~ms} ^ { - 1 }\). When the string makes an angle of \(\theta\) with the downward vertical the particle has speed \(v \mathrm {~ms} ^ { - 1 }\) (see diagram).

1. Show that, while the string is taut, the tension, \(T \mathrm {~N}\), in the string is given by $$T = 5.88 \cos \theta + 0.4 u ^ { 2 } - 3.92 .$$
2. Find the least value of \(u\) for which the particle will move in a complete circle.
3. If in fact \(u = 3.5 \mathrm {~ms} ^ { - 1 }\), find the speed of the particle at the point where the string first becomes slack.

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. Fig. 2 shows a car of mass 800 kg moving at constant speed in a horizontal circle with centre C and radius 45 m , on a road which is banked at an angle of \(18 ^ { \circ }\) to the horizontal. The forces shown are the weight \(W\) of the car, the normal reaction, \(R\), of the road on the car and the frictional force \(F\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{86dd0c01-970d-4b67-9a6c-5df276a4a2be-3_286_970_402_561} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure}
   1. Given that the frictional force is zero, find the speed of the car.
   2. Given instead that the speed of the car is \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), find the frictional force and the normal reaction.
2. One end of a light inextensible string is attached to a fixed point O , and the other end is attached to a particle P of mass \(m \mathrm {~kg}\). Starting with the string taut and P vertically below \(\mathrm { O } , \mathrm { P }\) is set in motion with a horizontal velocity of \(7 \mathrm {~ms} ^ { - 1 }\). It then moves in part of a vertical circle with centre O . The string becomes slack when the speed of P is \(2.8 \mathrm {~ms} ^ { - 1 }\). Find the length of the string. Find also the angle that OP makes with the upward vertical at the instant when the string becomes slack.

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.

5. A particle \(P\), of mass \(m \mathrm {~kg}\), is attached to one end of a light inextensible string of length \(l \mathrm {~m}\). The other end of the string is attached to a fixed point \(O\). Initially, \(P\) is held at rest with the string just taut and making an angle of \(60 ^ { \circ }\) with the downward vertical. It is then given a velocity \(u \mathrm {~ms} ^ { - 1 }\) perpendicular to the string in a downward direction.

1. 1. When the string makes an angle \(\theta\) with the downward vertical, the velocity of the particle is \(v\) and the tension in the string is \(T\). Find an expression for \(T\) in terms of \(m , l , u ^ { 2 }\) and \(\theta\).
   2. Given that \(P\) describes complete circles in the subsequent motion, show that \(u ^ { 2 } > 4 l g\).
2. Given that now \(u ^ { 2 } = 3 l g\), find the position of the string when circular motion ceases. Briefly describe the motion of \(P\) after circular motion has ceased.
3. The string is replaced by a light rigid rod. Given that \(P\) describes complete circles in the subsequent motion, show that \(u ^ { 2 } > k l g\), where \(k\) is to be determined.

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.

7 A small bead, of mass \(m\), is suspended from a fixed point \(O\) by a light inextensible string, of length \(a\). The bead is then set into circular motion with the string taut at \(B\), where \(B\) is vertically below \(O\), with a horizontal speed \(u\).
\includegraphics[max width=\textwidth, alt={}, center]{03994596-21ad-4201-8d64-ba2d7b7e0a77-5_451_458_461_760}

1. Given that the string does not become slack, show that the least value of \(u\) required for the bead to make complete revolutions about \(O\) is \(\sqrt { 5 a g }\).
2. In the case where \(u = \sqrt { 5 a g }\), find, in terms of \(g\) and \(m\), the tension in the string when the bead is at the point \(C\), which is at the same horizontal level as \(O\), as shown in the diagram.
3. State one modelling assumption that you have made in your solution.

1
\includegraphics[max width=\textwidth, alt={}, center]{d6a0d7a6-4166-4c26-a461-39b2414c0412-02_494_390_251_255} A smooth wire is shaped into a circle of radius 2.5 m which is fixed in a vertical plane with its centre at a point \(O\). A small bead \(B\) is threaded onto the wire. \(B\) is held with \(O B\) vertical and is then projected horizontally with an initial speed of \(8.4 \mathrm {~ms} ^ { - 1 }\) (see diagram).

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