Conical pendulum – horizontal circle in free space (no surface)

\includegraphics{figure_3} One end of a light inextensible string is attached to a fixed point \(A\) and the other end of the string is attached to a particle \(P\). The particle \(P\) moves with constant angular speed \(5\) rad s\(^{-1}\) in a horizontal circle which has its centre \(O\) vertically below \(A\). The string makes an angle \(\theta\) with the vertical (see diagram). The tension in the string is three times the weight of \(P\).

1. Show that the length of the string is \(1.2\) m. [3]
2. Find the speed of \(P\). [4]

A particle \(P\) 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 \(O\). The particle moves in a horizontal circle of radius \(0.8\) m with the string making a constant angle of \(60°\) with the vertical. Calculate the speed of the particle and the tension in the string. [6]

One end of a light inextensible string of length \(a\) is attached to a fixed point \(O\). The other end of the string is attached to a particle of mass \(m\). The string is taut and makes an angle \(\theta\) with the downward vertical through \(O\), where \(\cos \theta = \frac{2}{3}\). The particle moves in a horizontal circle with speed \(v\). Find \(v\) in terms of \(a\) and \(g\). [4]

A particle 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 \(A\). The particle moves with constant angular speed \(\omega\) in a horizontal circle. The centre of the circle is vertically below \(A\) and the radius of the circle is \(r\). Show that \(\omega^2 = \frac{g}{\sqrt{l^2 - r^2}}\) [8]

\includegraphics{figure_1} A garden game is played with a small ball \(B\) of mass \(m\) attached to one end of a light inextensible string of length \(13l\). The other end of the string is fixed to a point \(A\) on a vertical pole as shown in Figure 1. The ball is hit and moves with constant speed in a horizontal circle of radius \(5l\) and centre \(C\), where \(C\) is vertically below \(A\). Modelling the ball as a particle, find

1. the tension in the string, [3]
2. the speed of the ball. [4]

A particle \(P\) of mass 0.5 kg is attached to one end of a light inextensible string of length 1.5 m. The other end of the string is attached to a fixed point \(A\). The particle is moving, with the string taut, in a horizontal circle with centre \(O\) vertically below \(A\). The particle is moving with constant angular speed 2.7 rad s\(^{-1}\). Find

1. the tension in the string, [4]
2. the angle, to the nearest degree, that \(AP\) makes with the downward vertical. [3]

\includegraphics{figure_1} A metal ball \(B\) of mass \(m\) is attached to one end of a light inextensible string. The other end of the string is attached to a fixed point \(A\). The ball \(B\) moves in a horizontal circle with centre \(O\) vertically below \(A\), as shown in Fig. 1. The string makes a constant angle \(\alpha°\) with the downward vertical and \(B\) moves with constant angular speed \(\sqrt{(2gk)}\), where \(k\) is a constant. The tension in the string is \(3mg\). By modelling \(B\) as a particle, find

1. the value of \(\alpha\), [4]
2. the length of the string. [5]

A light inextensible string of length \(l\) has one end attached to a fixed point \(A\). The other end is attached to a particle \(P\) of mass \(m\). The particle moves with constant speed \(v\) in a horizontal circle with the string taut. The centre of the circle is vertically below \(A\) and the radius of the circle is \(r\). Show that $$gr^2 = v^2\sqrt{l^2 - r^2}.$$ [9]

A particle, of mass 0.8 kg, is attached to one end of a light inextensible string. The other end of the string is attached to the fixed point \(O\). The particle is set in motion, so that it moves in a horizontal circle at constant speed, with the string at an angle of \(35°\) to the vertical. The centre of this circle is vertically below \(O\), as shown in the diagram. \includegraphics{figure_4} The particle moves in a horizontal circle and completes 20 revolutions each minute.

1. Find the angular speed of the particle in radians per second. [2 marks]
2. Find the tension in the string. [3 marks]
3. Find the radius of the horizontal circle. [4 marks]

A particle \(P\) of mass \(m\) kg moves in a horizontal circle at one end of a light inextensible string of length 40 cm, as shown. The other end of the string is attached to a fixed point \(O\). The angular velocity of \(P\) is \(\omega\) rad s\(^{-1}\). \includegraphics{figure_2} If the angle \(\theta\) which the string makes with the vertical must not exceed 60°, calculate the greatest possible value of \(\omega\). [7 marks]

A conical pendulum consists of a light string and a particle of mass \(m\) kg The conical pendulum completes horizontal circles with radius \(r\) metres and angular speed \(\omega\) radians per second. The string makes an angle \(\theta\) with the downward vertical. The tension in the string is \(T\) newtons. The conical pendulum and the forces acting on the particle are shown in the diagram. \includegraphics{figure_3} Which one of the following statements is correct? Tick (\(\checkmark\)) one box. [1 mark] \(T \cos \theta = mr\omega^2\) \quad \(\square\) \(T \sin \theta = mr\omega^2\) \quad \(\square\) \(T \cos \theta = \frac{m\omega^2}{r}\) \quad \(\square\) \(T \sin \theta = \frac{m\omega^2}{r}\) \quad \(\square\)

\includegraphics{figure_4} The diagram shows a particle P, of mass 0.1 kg, which is attached by a light inextensible string of length 0.5 m to a fixed point O. P moves with constant angular speed 5 rad s\(^{-1}\) in a horizontal circle with centre vertically below O. The string is inclined at an angle \(\theta\) to the vertical.

1. Determine the tension in the string. [3]
2. Find the value of \(\theta\). [2]
3. Find the kinetic energy of P. [2]

\includegraphics{figure_3} The diagram shows a particle P, of mass 0.2 kg, which is attached by a light inextensible string of length 0.75 m to a fixed point O. Particle P moves with constant angular speed \(\omega \text{ rad s}^{-1}\) in a horizontal circle with centre vertically below O. The string is inclined at an angle \(\theta\) to the vertical. The greatest tension that the string can withstand without breaking is 15 N.

1. Find the greatest possible value of \(\theta\), giving your answer to the nearest degree. [2]
2. Determine the greatest possible value of \(\omega\). [3]

A particle of mass \(m\) kg is attached to one end of a light inextensible string. The other end of the string is attached to a fixed point \(O\). The particle is set in motion such that it moves in a horizontal circle of radius 2 m with constant speed 4.8 ms\(^{-1}\). Calculate the angle the string makes with the vertical. [6]