Elastic string – conical pendulum (string inclined to vertical)

4 \includegraphics[max width=\textwidth, alt={}, center]{b8e52188-f9a6-46fc-90bf-97965c6dd324-07_337_526_262_726} \includegraphics[max width=\textwidth, alt={}, center]{b8e52188-f9a6-46fc-90bf-97965c6dd324-07_111_116_486_1308} A particle \(P\) of mass 0.3 kg is attached to a fixed point \(A\) by a light elastic string of natural length 0.8 m and modulus of elasticity 16 N . The particle \(P\) moves in a horizontal circle which has centre \(O\). It is given that \(A O\) is vertical and that angle \(O A P\) is \(60 ^ { \circ }\) (see diagram). Calculate the speed of \(P\). [6]

4. A light elastic string \(A B\) has natural length 0.8 m and modulus of elasticity 19.6 N . The end \(A\) is attached to a fixed point. A particle of mass 0.5 kg is attached to the end \(B\). The particle is moving with constant angular speed \(\omega\) rad s \(^ { - 1 }\) in a horizontal circle whose centre is vertically below \(A\). The string is inclined at \(60 ^ { \circ }\) to the vertical.

1. Show that the extension of the string is 0.4 m .
2. Find the value of \(\omega\).

4. \begin{figure}[h] \end{figure} A particle \(P\) of mass \(m\) is attached to one end of a light elastic string, of natural length \(2 a\) and modulus of elasticity \(6 m g\). The other end of the string is attached to a fixed point \(A\). The particle moves with constant speed \(v\) in a horizontal circle with centre \(O\), where \(O\) is vertically below \(A\) and \(O A = 2 a\), as shown in Figure 2 .

1. Show that the extension in the string is \(\frac { 2 } { 5 } a\).
2. Find \(v ^ { 2 }\) in terms of \(a\) and \(g\).

2. A light elastic string has natural length 1.2 m and modulus of elasticity \(\lambda\) newtons. One end of the string is attached to a fixed point \(O\). A particle of mass 0.5 kg is attached to the other end of the string. The particle is moving with constant angular speed \(\omega \mathrm { rad } \mathrm { s } ^ { - 1 }\) in a horizontal circle with the string stretched. The circle has radius 0.9 m and its centre is vertically below \(O\). The string is inclined at \(60 ^ { \circ }\) to the horizontal. Find

1. the value of \(\lambda\),
2. the value of \(\omega\).

A particle \(P\) of mass \(0.15\) kg is attached to one end of a light elastic string of natural length \(0.4\) m and modulus of elasticity \(12\) N. The other end of the string is attached to a fixed point \(A\). The particle \(P\) moves in a horizontal circle which has its centre vertically below \(A\), with the string inclined at \(\theta°\) to the vertical and \(AP = 0.5\) m.

1. Find the angular speed of \(P\) and the value of \(\theta\). [5]
2. Calculate the difference between the elastic potential energy stored in the string and the kinetic energy of \(P\). [4]

A particle \(P\) of mass \(0.15\) kg is attached to one end of a light elastic string of natural length \(0.4\) m and modulus of elasticity \(12\) N. The other end of the string is attached to a fixed point \(A\). The particle \(P\) moves in a horizontal circle which has its centre vertically below \(A\), with the string inclined at \(θ°\) to the vertical and \(AP = 0.5\) m.

1. Find the angular speed of \(P\) and the value of \(θ\). [5]
2. Calculate the difference between the elastic potential energy stored in the string and the kinetic energy of \(P\). [4]

\includegraphics{figure_7} One end of a light elastic string with modulus of elasticity \(15\) N is attached to a fixed point \(A\) which is \(2\) m vertically above a fixed small smooth ring \(R\). The string has natural length \(2\) m and it passes through \(R\). The other end of the string is attached to a particle \(P\) of mass \(m\) kg which moves with constant angular speed \(\omega\) rad s\(^{-1}\) in a horizontal circle which has its centre \(0.4\) m vertically below the ring. \(PR\) makes an acute angle \(\theta\) with the vertical (see diagram).

1. Show that the tension in the string is \(\frac{3}{\cos\theta}\) N and hence find the value of \(m\). [4]
2. Show that the value of \(\omega\) does not depend on \(\theta\). [4]

It is given that for one value of \(\theta\) the elastic potential energy stored in the string is twice the kinetic energy of \(P\).

1. Find this value of \(\theta\). [4]

One end of a light elastic string, of natural length \(12a\) and modulus of elasticity \(kmg\), is attached to a fixed point \(O\). The other end of the string is attached to a particle of mass \(m\). The particle moves with constant speed \(\frac{2}{3}\sqrt{3ag}\) in a horizontal circle with centre at a distance \(12a\) below \(O\). The string is inclined at an angle \(\theta\) to the downward vertical through \(O\).

1. Find, in terms of \(a\), the extension of the string. [5]
2. Find the value of \(k\). [3]

One end of a light elastic string, of natural length \(a\) and modulus of elasticity \(4mg\), is attached to a fixed point \(O\). The other end of the string is attached to a particle of mass \(m\). The particle moves in a horizontal circle with a constant angular speed \(\sqrt{\frac{g}{a}}\) with the string inclined at an angle \(\theta\) to the downward vertical through \(O\). The length of the string during this motion is \((k+1)a\).

1. Find the value of \(k\). [4]
2. Find the value of \(\cos\theta\). [2]

A particle \(P\) of mass \(m\) kg moves in a horizontal circle at one end of a light elastic string of natural length \(l\) m and modulus of elasticity \(mg\) N. The other end of the string is attached to a fixed point \(O\). Given that the string makes an angle of \(60°\) with the vertical,

1. show that \(OP = 3l\) m. [4 marks]
2. Find, in terms of \(l\) and \(g\), the angular speed of \(P\). [4 marks]

In this question use \(g = 9.81 \text{ m s}^{-2}\) A conical pendulum is made from an elastic string and a sphere of mass 0.2 kg The string has natural length 1.6 metres and modulus of elasticity 200 N The sphere describes a horizontal circle of radius 0.5 metres at a speed of \(v \text{ m s}^{-1}\) The angle between the elastic string and the vertical is \(\alpha\)

1. Show that $$62.5 - 200 \sin \alpha = 1.962 \tan \alpha$$ [5 marks]
2. Use your calculator to find \(\alpha\) [1 mark]
3. Find the value of \(v\) [4 marks]