Particle in circular motion with string/rod

\includegraphics{figure_6} The diagram shows a smooth narrow tube formed into a fixed vertical circle with centre \(O\) and radius 0.9 m. A light elastic string with modulus of elasticity 8 N and natural length 1.2 m has one end attached to the highest point \(A\) on the inside of the tube. The other end of the string is attached to a particle \(P\) of mass 0.2 kg. The particle is released from rest at the lowest point on the inside of the tube. By considering energy, calculate

1. the speed of \(P\) when it is at the same horizontal level as \(O\), [4]
2. the speed of \(P\) at the instant when the string becomes slack. [3]

\includegraphics{figure_7} A particle \(P\) of mass \(m\) is attached to one end of a light rod of length \(3a\). The other end of the rod is able to pivot smoothly about the fixed point \(A\). The particle is also attached to one end of a light spring of natural length \(a\) and modulus of elasticity \(kmg\). The other end of the spring is attached to a fixed point \(B\). The points \(A\) and \(B\) are in a horizontal line, a distance \(5a\) apart, and these two points and the rod are in a vertical plane. Initially, \(P\) is held in equilibrium by a vertical force \(F\) with the stretched length of the spring equal to \(4a\) (see diagram). The particle is released from rest in this position and has a speed of \(\frac{6}{5}\sqrt{2ag}\) when the rod becomes horizontal.

1. Find the value of \(k\). [5]
2. Find \(F\) in terms of \(m\) and \(g\). [2]
3. Find, in terms of \(m\) and \(g\), the tension in the rod immediately before it is released. [2]

\includegraphics{figure_6} A particle \(P\) of weight \(6\) N is attached to the highest point \(A\) of a fixed smooth sphere by a light elastic string. The sphere has centre \(O\) and radius \(0.8\) m. The string has natural length \(\frac{1}{10}\pi\) m and modulus of elasticity \(9\) N. \(P\) is released from rest at a point \(X\) on the sphere where \(OX\) makes an angle of \(\frac{1}{3}\pi\) radians with the upwards vertical. \(P\) remains in contact with the sphere as it moves upwards to \(A\). At time \(t\) seconds after the release, \(OP\) makes an angle of \(\theta\) radians with the upwards vertical (see diagram). When \(\theta = \frac{1}{4}\pi\), \(P\) passes through the point \(Y\).

1. Show that as \(P\) moves from \(X\) to \(Y\) its gravitational potential energy increases by \(2.4(\sqrt{3} - \sqrt{2})\) J and the elastic potential energy in the string decreases by \(0.4\pi\) J. [5]
2. Verify that the transverse acceleration of \(P\) is zero when \(\theta = \frac{1}{4}\pi\), and hence find the maximum speed of \(P\). [6]

\includegraphics{figure_1} A smooth wire \(PMQ\) is in the shape of a semicircle with centre \(O\) and radius \(a\). The wire is fixed in a vertical plane with \(PQ\) horizontal and the mid-point \(M\) of the wire vertically below \(O\). A smooth bead \(B\) of mass \(m\) is threaded on the wire and is attached to one end of a light elastic string. The string has modulus of elasticity \(4mg\) and natural length \(\frac{3}{4}a\). The other end of the string is attached to a fixed point \(P\) which is a distance \(a\) vertically above \(O\), as shown in Fig. 1.

1. Show that, when \(\angle BFO = \theta\), the potential energy of the system is $$\frac{1}{16}mga(8 \cos \theta - 5)^2 - 2mga \cos^2\theta + \text{constant}.$$ [6]
2. Hence find the values of \(\theta\) for which the system is in equilibrium. [6]
3. Determine the nature of the equilibrium at each of these positions. [5]