Smooth ring on rotating string

4. \begin{figure}[h] \end{figure} A small smooth bead \(P\) is threaded on a light inextensible string of length \(8 a\). One end of the string is attached to a fixed point \(A\) on a smooth horizontal table. The other end of the string is attached to the fixed point \(B\), where \(B\) is vertically above \(A\) and \(A B = 4 a\), as shown in Figure 2. The bead moves with constant angular speed, in a horizontal circle, centre \(A\), with \(A P\) horizontal. The bead remains in contact with the table and both parts of the string, \(A P\) and \(B P\), are taut. The time for \(P\) to complete one revolution is \(S\). Show that \(\quad S \geqslant \pi \sqrt { \frac { 6 a } { g } }\)

5. \begin{figure}[h] \end{figure} A small smooth ring \(R\) of mass \(m\) is threaded on to a thin smooth fixed vertical pole. One end of a light inextensible string of length \(2 l\) is attached to a point \(A\) on the pole. The other end of the string is attached to \(R\). A particle \(P\) of mass \(m\) is attached to the midpoint of the string. The particle \(P\) moves with constant angular speed in a horizontal circle, with both halves of the string taut, and \(A R = \frac { 6 l } { 5 }\), as shown in Figure 2. It may be assumed that in this motion the string does not wrap itself around the pole and that at any instant, the triangle \(A P R\) lies in a vertical plane.

1. Show that the tension in the lower half of the string is \(\frac { 5 m g } { 3 }\)
2. Find, in terms of \(l\) and \(g\), the time for \(P\) to complete one revolution.

5. \begin{figure}[h] \end{figure} One end of a light inextensible string is attached to a fixed point \(A\). The other end of the string is attached to a fixed point \(B\), vertically below \(A\), where \(A B = h\). A small smooth ring \(R\) of mass \(m\) is threaded on the string. The ring \(R\) moves in a horizontal circle with centre \(B\), as shown in Figure 3. The upper section of the string makes a constant angle \(\theta\) with the downward vertical and \(R\) moves with constant angular speed \(\omega\). The ring is modelled as a particle.

1. Show that \(\omega ^ { 2 } = \frac { g } { h } \left( \frac { 1 + \sin \theta } { \sin \theta } \right)\).
2. Deduce that \(\omega > \sqrt { \frac { 2 g } { h } }\). Given that \(\omega = \sqrt { \frac { 3 g } { h } }\),
3. find, in terms of \(m\) and \(g\), the tension in the string.

2. \begin{figure}[h] \end{figure} A light inextensible string of length \(8 l\) has its ends fixed to two points \(A\) and \(B\), where \(A\) is vertically above \(B\). A small smooth ring of mass \(m\) is threaded on the string. The ring is moving with constant speed in a horizontal circle with centre \(B\) and radius 3l, as shown in Fig. 2. Find

1. the tension in the string,
2. the speed of the ring.
3. State briefly in what way your solution might no longer be valid if the ring were firmly attached to the string.
   (1) \section*{3.} \section*{Figure 3}
   \includegraphics[max width=\textwidth, alt={}]{044c5866-0a12-4309-8ced-b463e1615fb0-3_564_1051_438_541}
   A child's toy consists of a uniform solid hemisphere attached to a uniform solid cylinder. The plane face of the hemisphere coincides with the plane face of the cylinder, as shown in Fig. 3. The cylinder and the hemisphere each have radius \(r\), and the height of the cylinder is \(h\). The material of the hemisphere is 6 times as dense as the material of the cylinder. The toy rests in equilibrium on a horizontal plane with the cylinder above the hemisphere and the axis of the cylinder vertical.

5. \begin{figure}[h] \end{figure} One end of a light inextensible string is attached to a fixed point \(A\). The other end of the string is attached to a fixed point \(B\), vertically below \(A\), where \(A B = h\). A small smooth ring \(R\) of mass \(m\) is threaded on the string. The ring \(R\) moves in a horizontal circle with centre \(B\), as shown in Figure 3. The upper section of the string makes a constant angle \(\theta\) with the downward vertical and \(R\) moves with constant angular speed \(\omega\). The ring is modelled as a particle.

1. Show that \(\omega ^ { 2 } = \frac { g } { h } \left( \frac { 1 + \sin \theta } { \sin \theta } \right)\).
2. Deduce that \(\omega > \sqrt { \frac { 2 g } { h } }\). Given that \(\omega = \sqrt { \frac { 3 g } { h } }\),
3. find, in terms of \(m\) and \(g\), the tension in the string.

3 \includegraphics[max width=\textwidth, alt={}, center]{35477eb8-59e0-4de6-889c-1f5841f65eec-2_451_533_1676_808} One end of a light inextensible string of length 1.6 m is attached to a point \(P\). The other end is attached to the point \(Q\), vertically below \(P\), where \(P Q = 0.8 \mathrm {~m}\). A small smooth bead \(B\), of mass 0.01 kg , is threaded on the string and moves in a horizontal circle, with centre \(Q\) and radius \(0.6 \mathrm {~m} . Q B\) rotates with constant angular speed \(\omega\) rad s \(^ { - 1 }\) (see diagram).

1. Show that the tension in the string is 0.1225 N .
2. Find \(\omega\).
3. Calculate the kinetic energy of the bead.

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

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

6. \begin{figure}[h] \end{figure} The two ends of a light inextensible string of length \(3 a\) are attached to fixed points \(Q\) and \(R\) which are a distance of \(a \sqrt { } 3\) apart with \(R\) vertically below \(Q\). A particle \(P\) of mass \(m\) is attached to the string at a distance of \(2 a\) from \(Q\). \(P\) is given a horizontal speed, \(u\), such that it moves in a horizontal circle with both sections of the string taut as shown in Figure 3.

1. Show that \(\angle P R Q\) is a right angle.
2. Find \(\angle P Q R\) in degrees.
3. Find, in terms of \(a , g , m\) and \(u\), the tension in the section of string
   1. \(P Q\),
   2. \(P R\).
4. Show that \(u ^ { 2 } \geq \frac { g a } { \sqrt { 3 } }\).

2. \begin{figure}[h] \end{figure} One end of a string of length \(3 a\) is attached to a point \(A\) and the other end is attached to a point \(B\) on a smooth horizontal table. The point \(B\) is vertically below \(A\) with \(A B = a \sqrt { 3 }\) A small smooth bead, \(P\), of mass \(m\) is threaded on to the string. The bead \(P\) moves on the table in a horizontal circle, with centre \(B\), with constant speed \(U\). Both portions, \(A P\) and \(B P\), of the string are taut, as shown in Figure 2. The string is modelled as being light and inextensible and the bead is modelled as a particle.

1. Show that \(A P = 2 a\)
2. Find, in terms of \(m , U\) and \(a\), the tension in the string.
3. Show that \(U ^ { 2 } < a g \sqrt { 3 }\)
4. Describe what would happen if \(U ^ { 2 } > a g \sqrt { 3 }\)
5. State briefly how the tension in the string would be affected if the string were not modelled as being light.

4. \begin{figure}[h] \end{figure} One end of a light inextensible string of length \(2 l\) is attached to a fixed point \(A\). A small smooth ring \(R\) of mass \(m\) is threaded on the string and the other end of the string is attached to a fixed point \(B\). The point \(B\) is vertically below \(A\), with \(A B = l\). The ring is then made to move with constant speed \(V\) in a horizontal circle with centre \(B\). The string is taut and \(B R\) is horizontal, as shown in Figure 4.

1. Show that \(B R = \frac { 31 } { 4 }\) Given that air resistance is negligible,
2. find, in terms of \(m\) and \(g\), the tension in the string,
3. find \(V\) in terms of \(g\) and \(l\).

4. \begin{figure}[h] \end{figure} A small smooth ring \(R\) of mass \(m\) is threaded onto a light inextensible string. One end of the string is attached to a fixed point \(A\) and the other end of the string is attached to the fixed point \(B\) such that \(B\) is vertically above \(A\) and \(A B = 6 a\) The ring moves with constant angular speed \(\omega\) in a horizontal circle with centre \(A\). The string is taut and \(B R\) makes a constant angle \(\theta\) with the downward vertical, as shown in Figure 2. The ring is modelled as a particle.
Given that \(\tan \theta = \frac { 8 } { 15 }\)

1. find, in terms of \(m\) and \(g\), the magnitude of the tension in the string,
2. find \(\omega\) in terms of \(a\) and \(g\)

7 \includegraphics[max width=\textwidth, alt={}, center]{86cc07e7-ea69-4480-96c8-82b818445199-3_599_499_1279_822} A light inextensible string of length \(4 a\) has one end fixed at a point \(P\) and the other end fixed at a point \(Q\), which is vertically below \(P\) and at a distance \(3 a\) from \(P\). A small smooth ring \(R\) of mass \(m\) is threaded on the string. \(R\) moves in a horizontal circle with centre \(Q\) and with the string taut (see diagram).

1. Show that \(Q R = \frac { 7 } { 8 } a\).
2. Find the speed of \(R\) in terms of \(a\) and \(g\).

9 \includegraphics[max width=\textwidth, alt={}, center]{09939c3a-7829-4784-8e6d-ee5356c22cd7-4_433_428_1219_863} A light inextensible string of length 1.4 m has its ends attached to two points \(A\) and \(C\), where \(A\) is 1 m vertically above \(C\). A smooth bead \(B\) of mass 0.2 kg is threaded on the string and rotates in a horizontal circle with the string taut. The distance \(B A\) is 0.8 m (see diagram). Find

1. the tension in the string,
2. the time taken for the bead to perform one complete circle.

\includegraphics{figure_6} A light inextensible string passes through a small smooth bead \(B\) of mass \(0.4 \text{ kg}\). One end of the string is attached to a fixed point \(A\) \(0.4 \text{ m}\) above a fixed point \(O\) on a smooth horizontal surface. The other end of the string is attached to a fixed point \(C\) which is vertically below \(A\) and \(0.3 \text{ m}\) above the surface. The bead moves with constant speed on the surface in a circle with centre \(O\) and radius \(0.3 \text{ m}\) (see diagram).

1. Given that the tension in the string is \(2 \text{ N}\), calculate
   1. the angular speed of the bead, [3]
   2. the magnitude of the contact force exerted on the bead by the surface. [2]
2. Given instead that the bead is about to lose contact with the surface, calculate the speed of the bead. [4]

\includegraphics{figure_6} A light inextensible string passes through a small smooth bead \(B\) of mass \(0.4\) kg. One end of the string is attached to a fixed point \(A\) \(0.4\) m above a fixed point \(O\) on a smooth horizontal surface. The other end of the string is attached to a fixed point \(C\) which is vertically below \(A\) and \(0.3\) m above the surface. The bead moves with constant speed on the surface in a circle with centre \(O\) and radius \(0.3\) m (see diagram).

1. Given that the tension in the string is \(2\) N, calculate
   1. the angular speed of the bead, [3]
   2. the magnitude of the contact force exerted on the bead by the surface. [2]
2. Given instead that the bead is about to lose contact with the surface, calculate the speed of the bead. [4]

A smooth bead \(B\) of mass 0.3 kg is threaded on a light inextensible string of length 0.9 m. One end of the string is attached to a fixed point \(A\), and the other end of the string is attached to a fixed point \(C\) which is vertically below \(A\). The tension in the string is 7 N, and the bead rotates with angular speed \(ω\) rad s\(^{-1}\) in a horizontal circle about the vertical axis through \(A\) and \(C\).

1. Given that \(B\) moves in a circle with centre \(C\) and radius 0.2 m, calculate \(ω\), and hence find the kinetic energy of \(B\). [5]
2. Given instead that angle \(ABC = 90°\), and that \(AB\) makes an angle \(\tan^{-1}(\frac{4}{3})\) with the vertical, calculate \(T\) and \(ω\). [6]

\includegraphics{figure_4} One end of a light inextensible string is attached to a fixed point \(A\). The string passes through a smooth bead \(B\) of mass \(0.3\text{ kg}\) and the other end of the string is attached to a fixed point \(C\) vertically below \(A\). The bead \(B\) moves with constant speed in a horizontal circle of radius \(0.6\text{ m}\) which has its centre between \(A\) and \(C\). The string makes an angle of \(30°\) with the vertical at \(A\) and an angle of \(45°\) with the vertical at \(C\) (see diagram).

1. Calculate the speed of \(B\). [5]

The lower end of the string is detached from \(C\), and \(B\) is now attached to this end of the string. The other end of the string remains attached to \(A\). The bead is set in motion so that it moves with angular speed \(3\text{ rad s}^{-1}\) in a horizontal circle which has its centre vertically below \(A\).

1. Calculate the tension in the string. [3]

\includegraphics{figure_4} One end of a light inextensible string is attached to a fixed point \(A\). The string passes through a smooth bead \(B\) of mass \(0.3\,\text{kg}\) and the other end of the string is attached to a fixed point \(C\) vertically below \(A\). The bead \(B\) moves with constant speed in a horizontal circle of radius \(0.6\,\text{m}\) which has its centre between \(A\) and \(C\). The string makes an angle of \(30°\) with the vertical at \(A\) and an angle of \(45°\) with the vertical at \(C\) (see diagram).

1. Calculate the speed of \(B\). [5]

The lower end of the string is detached from \(C\), and \(B\) is now attached to this end of the string. The other end of the string remains attached to \(A\). The bead is set in motion so that it moves with angular speed \(3\,\text{rad s}^{-1}\) in a horizontal circle which has its centre vertically below \(A\).

1. Calculate the tension in the string. [3]

\includegraphics{figure_2} A light inextensible string of length \(a\) is threaded through a fixed smooth ring \(R\). One end of the string is attached to a particle \(A\) of mass \(3m\). The other end of the string is attached to a particle \(B\) of mass \(m\). The particle \(A\) hangs in equilibrium at a distance \(x\) vertically below the ring. The angle between \(AR\) and \(BR\) is \(\theta\) (see diagram). The particle \(B\) moves in a horizontal circle with constant angular speed \(2\sqrt{\frac{g}{a}}\). Show that \(\cos \theta = \frac{1}{3}\) and find \(x\) in terms of \(a\). [5]

\includegraphics{figure_3} One end of a light inextensible string of length 1.6 m is attached to a point \(P\). The other end is attached to the point \(Q\), vertically below \(P\), where \(PQ = 0.8\) m. A small smooth bead \(B\), of mass 0.01 kg, is threaded on the string and moves in a horizontal circle, with centre \(Q\) and radius 0.6 m. \(QB\) rotates with constant angular speed \(\omega\) rad s\(^{-1}\) (see diagram).

1. Show that the tension in the string is 0.1225 N. [3]
2. Find \(\omega\). [3]
3. Calculate the kinetic energy of the bead. [2]

A light inelastic string of length \(l\) m passes through a small smooth ring which is fixed at a point \(O\) and is free to rotate about a vertical axis through \(O\). Particles \(P\) and \(Q\), of masses 0.06 kg and 0.04 kg respectively, are attached to the ends of the string.

1. \(Q\) describes a horizontal circle with centre \(P\), while \(P\) hangs at rest at a depth \(d\) m below \(O\). Show that \(d = \frac{2l}{5}\). [6 marks]
2. \(P\) and \(Q\) now both move in horizontal circles with the same angular velocity \(\omega\) rad s\(^{-1}\) about a vertical axis through \(O\). Show that \(OQ = \frac{3l}{5}\) m. [7 marks]

\includegraphics{figure_5}

\includegraphics{figure_12} The ends of a light inextensible string are fixed to two points A and B in the same vertical line, with A above B. The string passes through a small smooth ring of mass \(m\). The ring is fastened to the string at a point P. When the string is taut the angle APB is a right angle, the angle BAP is \(\theta\) and the perpendicular distance of P from AB is \(r\). The ring moves in a horizontal circle with constant angular velocity \(\omega\) and the string taut as shown in Fig. 12.

1. By resolving horizontally and vertically, show that the tension in the part of the string BP is \(m(r\omega^2\cos\theta - g\sin\theta)\). [6]
2. Find a similar expression, in terms of \(r\), \(\omega\), \(m\), \(g\) and \(\theta\), for the tension in the part of the string AP. [2]

It is given that AB = 5a and AP = 4a.

1. Show that \(16a\omega^2 > 5g\). [3]

The ring is now free to move on the string but remains in the same position on the string as before. The string remains taut and the ring continues to move in a horizontal circle.

1. Find the period of the motion of the ring, giving your answer in terms of \(a\), \(g\) and \(\pi\). [5]

Fig. 5 shows a light inextensible string of length 3.3 m passing through a small smooth ring R. The ends of the string are attached to fixed points A and B, where A is vertically above B. The ring R has mass 0.27 kg and is moving with constant speed in a horizontal circle of radius 1.2 m. The distances AR and BR are 2 m and 1.3 m respectively. \includegraphics{figure_5}

1. Show that the tension in the string is 6.37 N. [4]
2. Find the speed of R. [4]

\includegraphics{figure_4} As shown in the diagram, \(AB\) is a long thin rod which is fixed vertically with \(A\) above \(B\). One end of a light inextensible string of length \(1\) m is attached to \(A\) and the other end is attached to a particle \(P\) of mass \(m_1\) kg. One end of another light inextensible string of length \(1\) m is also attached to \(P\). Its other end is attached to a small smooth ring \(R\), of mass \(m_2\) kg, which is free to move on \(AB\). Initially, \(P\) moves in a horizontal circle of radius \(0.6\) m with constant angular velocity \(\omega\) rad s\(^{-1}\). The magnitude of the tension in string \(AP\) is denoted by \(T_1\) N while that in string \(PR\) is denoted by \(T_2\) N.

1. By considering forces on \(R\), express \(T_2\) in terms of \(m_2\). [2]
2. Show that
   1. \(T_1 = \frac{4g}{5}(m_1 + m_2)\). [2]
   2. \(\omega^2 = \frac{4g(m_1 + 2m_2)}{4m_1}\). [3]
3. Deduce that, in the case where \(m_1\) is much bigger than \(m_2\), \(\omega \approx 3.5\). [2]

In a different case, where \(m_1 = 2.5\) and \(m_2 = 2.8\), \(P\) slows down. Eventually the system comes to rest with \(P\) and \(R\) hanging in equilibrium.

1. Find the total energy lost by \(P\) and \(R\) as the angular velocity of \(P\) changes from the initial value of \(\omega\) rad s\(^{-1}\) to zero. [5]