Particle on table with string above

7 One end of a light inextensible string of length 0.15 m is attached to a fixed point which is above a smooth horizontal surface. A particle of mass 0.5 kg is attached to the other end of the string. The particle moves with constant speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a horizontal circle, with the string taut and making an angle of \(\theta ^ { \circ }\) with the downward vertical.

1. Given that \(\theta = 60\) and that the particle is not in contact with the surface, find \(v\).
2. Given instead that \(\theta = 45\) and \(v = 0.9\), and that the particle is in contact with the surface, find
   1. the tension in the string,
   2. the force exerted by the surface on the particle.

6 \includegraphics[max width=\textwidth, alt={}, center]{d6cb7a28-e8d7-4239-b9d3-120a284d7353-3_259_890_584_630} One end of a light inextensible string of length 0.2 m is attached to a fixed point \(A\) which is above a smooth horizontal table. A particle \(P\) of mass 0.3 kg is attached to the other end of the string. \(P\) moves on the table in a horizontal circle, with the string taut and making an angle of \(60 ^ { \circ }\) with the downward vertical (see diagram).

1. Calculate the tension in the string if the speed of \(P\) is \(1.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
2. For the motion as described, show that the angular speed of \(P\) cannot exceed \(10 \mathrm { rad } \mathrm { s } ^ { - 1 }\), and hence find the greatest possible value for the kinetic energy of \(P\).

6 A particle \(P\) of mass 0.6 kg is released from rest at a point above ground level and falls vertically. The motion of \(P\) is opposed by a force of magnitude \(3 v \mathrm {~N}\), where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the speed of \(P\). Immediately before \(P\) reaches the ground, \(v = 1.95\).

1. Calculate the time after its release when \(P\) reaches the ground. \(P\) is now projected horizontally with speed \(1.95 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) across a smooth horizontal surface. The motion of \(P\) is again opposed by a force of magnitude \(3 v \mathrm {~N}\), where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the speed of \(P\).
2. Calculate the distance \(P\) travels after projection before coming to rest.

3 A particle \(P\) of mass 0.4 kg is attached to a fixed point \(A\) by a light inextensible string of length 0.5 m . The point \(A\) is 0.3 m above a smooth horizontal surface. The particle \(P\) moves in a horizontal circle on the surface with constant angular speed \(5 \mathrm { rad } \mathrm { s } ^ { - 1 }\).

1. Calculate the tension in the string. \includegraphics[max width=\textwidth, alt={}, center]{f3a35846-075d-4e03-ba6b-82774ef0e4f8-05_67_1569_486_328}
2. Find the magnitude of the force exerted by the surface on \(P\).

3 \includegraphics[max width=\textwidth, alt={}, center]{0cb05368-9ddf-4564-8428-725c77193a1e-2_892_412_1217_865} A hollow cylinder of radius 0.35 m has a smooth inner surface. The cylinder is fixed with its axis vertical. One end of a light inextensible string of length 1.25 m is attached to a fixed point \(O\) on the axis of the cylinder. A particle \(P\) of mass 0.24 kg is attached to the other end of the string. \(P\) moves with constant speed in a horizontal circle, in contact with the inner surface of the cylinder, and with the string taut (see diagram).

1. Find the tension in the string.
2. Given that the magnitude of the acceleration of \(P\) is \(8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), find the force exerted on \(P\) by the cylinder.

2 \includegraphics[max width=\textwidth, alt={}, center]{b9080e9f-2c23-43ce-b171-bd68648dc56b-2_496_609_1535_769} One end of a light inextensible string of length 0.16 m is attached to a fixed point \(A\) which is above a smooth horizontal table. A particle \(P\) of mass 0.4 kg is attached to the other end of the string. \(P\) moves on the table in a horizontal circle, with the string taut and making an angle of \(30 ^ { \circ }\) with the downward vertical through \(A\) (see diagram). \(P\) moves with constant speed \(0.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find

1. the tension in the string,
2. the force exerted by the table on \(P\).

6 \includegraphics[max width=\textwidth, alt={}, center]{68acf474-5da2-4949-b3b2-fc42cd73bd4a-3_504_878_1557_632} One end of a light inextensible string of length 0.7 m is attached to a fixed point \(A\). The other end of the string is attached to a particle \(P\) of mass 0.25 kg . The particle \(P\) moves in a circle on a smooth horizontal table with constant speed \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The string is taut and makes an angle of \(40 ^ { \circ }\) with the vertical (see diagram). Find

1. the tension in the string,
2. the force exerted on \(P\) by the table. \(P\) now moves in the same horizontal circle with constant angular speed \(\omega \operatorname { rad~s } ^ { - 1 }\).
3. Find the maximum value of \(\omega\) for which \(P\) remains on the table.

3 \includegraphics[max width=\textwidth, alt={}, center]{af7d1fc8-5552-48b8-a359-895b2b5d3d6c-2_279_905_1560_621} One end of a light inextensible string of length 0.2 m is attached to a fixed point \(A\) which is above a smooth horizontal surface. A particle \(P\) of mass 0.6 kg is attached to the other end of the string. \(P\) moves in a circle on the surface with constant speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), with the string taut and making an angle of \(30 ^ { \circ }\) to the horizontal (see diagram).

1. Given that \(v = 1.5\), calculate the magnitude of the force that the surface exerts on \(P\).
2. Given instead that \(P\) moves with its greatest possible speed while remaining in contact with the surface, find \(v\).

5 A small ball \(B\) of mass 0.4 kg moves in a horizontal circle with centre \(O\) and radius 0.6 m on a smooth horizontal surface. One end of a light inextensible string is attached to \(B\); the other end of the string is attached to a fixed point 0.45 m vertically above \(O\).

1. Given that the tension in the string is 5 N , calculate the speed of \(B\).
2. Find the greatest possible tension in the string for the motion, and the corresponding angular speed of \(B\).

3 \includegraphics[max width=\textwidth, alt={}, center]{8a7016eb-4e76-4104-aa00-fbf09e1d739a-04_305_510_264_813} One end of a light inextensible string of length 0.4 m is attached to a fixed point \(A\) which is above a smooth horizontal surface. A particle \(P\) of mass 0.6 kg is attached to the other end of the string. \(P\) moves in a circle on the surface with constant speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), with the string taut and making an angle of \(60 ^ { \circ }\) with the horizontal (see diagram).

1. Given that \(v = 0.5\), calculate the magnitude of the force that the surface exerts on \(P\).
2. Find the greatest possible value of \(v\) for which \(P\) remains in contact with the surface.

3 \includegraphics[max width=\textwidth, alt={}, center]{2e4e6e32-eafc-4196-aaa8-42909cc2078e-04_305_510_264_813} One end of a light inextensible string of length 0.4 m is attached to a fixed point \(A\) which is above a smooth horizontal surface. A particle \(P\) of mass 0.6 kg is attached to the other end of the string. \(P\) moves in a circle on the surface with constant speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), with the string taut and making an angle of \(60 ^ { \circ }\) with the horizontal (see diagram).

1. Given that \(v = 0.5\), calculate the magnitude of the force that the surface exerts on \(P\).
2. Find the greatest possible value of \(v\) for which \(P\) remains in contact with the surface.

2. \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 point \(A\) which lies above a smooth horizontal table. The particle \(P\) moves in a horizontal circle on the table with the string taut. The centre of the circle is the point \(O\) on the table, where \(A O\) is vertical and the string makes a constant angle \(\theta ^ { \circ }\) with \(A O\), as shown in Figure 2. Given that \(P\) moves with constant angular speed \(\sqrt { \frac { 2 g } { a } }\), find the range of possible values of \(\theta\)

2. \begin{figure}[h] \end{figure} A particle \(P\) of mass \(m\) is attached to one end of a light inextensible string of length \(2 l\). The other end of the string is attached to a fixed point \(A\) above a smooth horizontal floor. The particle moves in a horizontal circle on the floor with the string taut. The centre \(O\) of the circle is vertically below \(A\) with \(O A = l\), as shown in Figure 2 . The particle moves with constant angular speed \(\omega\) and remains in contact with the floor.
Show that $$\omega \leqslant \sqrt { \frac { g } { l } }$$

7. \begin{figure}[h] \end{figure} A particle \(P\) of mass \(m\) is attached to one end of a light inextensible string of length \(2 a\). The other end of the string is fixed to a point \(A\) which is vertically above the point \(O\) on a smooth horizontal table. The particle \(P\) remains in contact with the surface of the table and moves in a circle with centre \(O\) and with angular speed \(\sqrt { \left( \frac { k g } { 3 a } \right) }\), where \(k\) is a constant. Throughout the motion the string remains taut and \(\angle A P O = 30 ^ { \circ }\), as shown in Figure 3.

1. Show that the tension in the string is \(\frac { 2 k m g } { 3 }\).
2. Find, in terms of \(m , g\) and \(k\), the normal reaction between \(P\) and the table.
3. Deduce the range of possible values of \(k\). The angular speed of \(P\) is changed to \(\sqrt { \left( \frac { 2 g } { a } \right) }\). The particle \(P\) now moves in a horizontal circle above the table. The centre of this circle is \(X\).
4. Show that \(X\) is the mid-point of \(O A\).

3. \begin{figure}[h] \end{figure} Figure 2 shows a particle \(B\), of mass \(m\), attached to one end of a light elastic string. The other end of the string is attached to a fixed point \(A\), at a distance \(h\) vertically above a smooth horizontal table. The particle moves on the table in a horizontal circle with centre \(O\), where \(O\) is vertically below \(A\). The string makes a constant angle \(\theta\) with the downward vertical and \(B\) moves with constant angular speed \(\omega\) about \(O A\).

1. Show that \(\omega ^ { 2 } \leqslant \frac { g } { h }\). The elastic string has natural length \(h\) and modulus of elasticity \(2 m g\).
   Given that \(\tan \theta = \frac { 3 } { 4 }\),
2. find \(\omega\) in terms of \(g\) and \(h\).

3. \begin{figure}[h] \end{figure} A particle \(P\) of mass \(m\) is attached to one end of a light inextensible string of length \(3 a\). The other end of the string is attached to a fixed point \(A\) which is a vertical distance \(a\) above a smooth horizontal table. The particle moves on the table in a circle whose centre \(O\) is vertically below \(A\), as shown in Fig. 1. The string is taut and the speed of \(P\) is \(2 \sqrt { } ( a g )\). Find

1. the tension in the string,
2. the normal reaction of the table on \(P\).

7 \includegraphics[max width=\textwidth, alt={}, center]{1fbb3693-0beb-47c8-800f-50041f105699-4_782_1006_274_571} One end of a light inextensible string of length 0.8 m is attached to a fixed point \(A\) which lies above a smooth horizontal table. The other end of the string is attached to a particle \(P\), of mass 0.3 kg , which moves in a horizontal circle on the table with constant angular speed \(2 \mathrm { rad } \mathrm { s } ^ { - 1 } . A P\) makes an angle of \(30 ^ { \circ }\) with the vertical (see diagram).

1. Calculate the tension in the string.
2. Calculate the normal contact force between the particle and the table. The particle now moves with constant speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and is on the point of leaving the surface of the table.
3. Calculate \(v\).

4 A particle \(P\) of mass 0.2 kg is attached to one end of a light inextensible string of length 1.2 m . The other end of the string is fixed at a point \(A\) which is 0.6 m above a smooth horizontal table. \(P\) moves on the table in a circular path whose centre \(O\) is vertically below \(A\).

1. Given that the angular speed of \(P\) is \(2.5 \mathrm { rad } \mathrm { s } ^ { - 1 }\), find
   1. the tension in the string,
   2. the normal reaction between the particle and the table.
   3. Find the greatest possible speed of \(P\), given that the particle remains in contact with the table.

2. \begin{figure}[h] \end{figure} A small smooth ring \(P\), of mass \(m\), is threaded onto a light inextensible string of length 4a. 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 a fixed point \(B\) which is vertically above \(A\). The ring moves in a horizontal circle with centre \(A\) and radius \(a\), as shown in Figure 2. The ring moves with constant angular speed \(\sqrt { \frac { 2 g } { 3 a } }\) about \(A B\).
The string remains taut throughout the motion.

1. Find, in terms of \(m\) and \(g\), the magnitude of the normal reaction between \(P\) and the table. The angular speed of \(P\) is now gradually increased.
2. Find, in terms of \(a\) and \(g\), the angular speed of \(P\) at the instant when it loses contact with the table.
3. Explain how you have used the fact that \(P\) is smooth.

5. \begin{figure}[h] \end{figure} A particle \(P\) of mass 0.75 kg is attached to one end of a light inextensible string of length 60 cm . The other end of the string is attached to a fixed point \(A\) that is vertically above the point \(O\) on a smooth horizontal table, such that \(O A = 40 \mathrm {~cm}\). The particle remains in contact with the table, with the string taut, and moves in a horizontal circle with centre \(O\), as shown in Figure 4. The particle is moving with a constant angular speed of 3 radians per second.

1. Find (i) the tension in the string,
   (ii) the normal reaction between \(P\) and the table. The angular speed of \(P\) is now gradually increased.
2. Find the angular speed of \(P\) at the instant \(P\) loses contact with the table.

6 \includegraphics[max width=\textwidth, alt={}, center]{a1a43547-0a68-4346-884a-0c6d9302cf24-4_547_597_251_735} A particle of mass 0.2 kg is attached to one end of a light inextensible string. The other end of the string is attached to a fixed point \(O\) which is 1.8 m above a smooth horizontal table. The particle moves on the table in a circular path at constant speed with the string taut (see diagram). The particle has a speed of \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and its angular velocity is \(0.625 \mathrm { rad } \mathrm { s } ^ { - 1 }\).

1. Show that the radius of the circular path is 0.8 m .
2. Find the magnitude of the normal contact force between the particle and the table. The speed is changed to \(v \mathrm {~ms} ^ { - 1 }\). At this speed the particle is just about to lose contact with the table.
3. Find the value of \(v\).

A small ball \(B\) of mass 0.4 kg moves in a horizontal circle with centre \(O\) and radius 0.6 m on a smooth horizontal surface. One end of a light inextensible string is attached to \(B\); the other end of the string is attached to a fixed point 0.45 m vertically above \(O\).

1. Given that the tension in the string is 5 N, calculate the speed of \(B\). [3]
2. Find the greatest possible tension in the string for the motion, and the corresponding angular speed of \(B\). [4]