The fixed points \(A\) and \(B\) are on a smooth horizontal surface with \(A B=2.6 \mathrm{~m}\). One end of a light elastic spring, of natural length 1.25 m and modulus of elasticity \(\lambda \mathrm{N}\), is attached to \(A\). The other end is attached to a particle \(P\) of mass 0.4 kg . One end of a second light elastic spring, of natural length 1.0 m and modulus of elasticity \(0.6 \lambda \mathrm{~N}\), is attached to \(B\); its other end is attached to \(P\). The system is in equilibrium with \(P\) on the surface at the point \(E\).
(i) Show that \(A E=1.4 \mathrm{~m}\).

The particle \(P\) is now displaced slightly from \(E\), along the line \(A B\).
(ii) Show that, in the subsequent motion, \(P\) performs simple harmonic motion.

(iii) Given that the period of the motion is \(\frac{1}{7} \pi \mathrm{~s}\), find the value of \(\lambda\).

Question 11 EITHER alternative.

The points \(A\) and \(B\) are a distance 1.2 m apart on a smooth horizontal surface. A particle \(P\) of mass \(\frac{2}{3}\ \mathrm{kg}\) is attached to one end of a light spring of natural length 0.6 m and modulus of elasticity 10 N. The other end of the spring is attached to the point \(A\). A second light spring, of natural length 0.4 m and modulus of elasticity 20 N, has one end attached to \(P\) and the other end attached to \(B\).

(i) Show that when \(P\) is in equilibrium \(AP=0.75\ \mathrm{m}\).

The particle \(P\) is displaced by 0.05 m from the equilibrium position towards \(A\) and then released from rest.

(ii) Show that \(P\) performs simple harmonic motion and state the period of the motion.

(iii) Find the speed of \(P\) when it passes through the equilibrium position.

(iv) Find the speed of \(P\) when its acceleration is equal to half of its maximum value.

Question 11 EITHER alternative.

A light spring has natural length \(a\) and modulus of elasticity \(k m g\). The spring lies on a smooth horizontal surface with one end attached to a fixed point \(O\). A particle \(P\) of mass \(m\) is attached to the other end of the spring. The system is in equilibrium with \(O P=a\). The particle is projected towards \(O\) with speed \(u\) and comes to instantaneous rest when \(O P=\frac{3}{4} a\).
(i) Use an energy method to show that \(k=\frac{16 u^{2}}{a g}\).
(ii) Show that \(P\) performs simple harmonic motion and find the period of this motion, giving your answer in terms of \(u\) and \(a\).
(iii) Find, in terms of \(u\) and \(a\), the time that elapses before \(P\) first loses 25% of its initial kinetic energy.

Question 11 EITHER alternative.

One end of a light elastic spring, of natural length \(0.8\ \mathrm{m}\) and modulus of elasticity \(40\ \mathrm{N}\), is attached to a fixed point \(O\). The spring hangs vertically, at rest, with particles of masses \(2\ \mathrm{kg}\) and \(M\ \mathrm{kg}\) attached to its free end. The \(M\ \mathrm{kg}\) particle becomes detached from the spring, and as a result the \(2\ \mathrm{kg}\) particle begins to move upwards.

(i) Show that the \(2\ \mathrm{kg}\) particle performs simple harmonic motion about its equilibrium position with period \(\frac25\pi\ \mathrm{s}\). State the distance below \(O\) of the centre of the oscillations.

The speed of the \(2\ \mathrm{kg}\) particle is \(0.4\ \mathrm{m\ s^{-1}}\) when its displacement from the centre of oscillation is \(0.06\ \mathrm{m}\).

(ii) Find the amplitude of the motion.

(iii) Deduce the value of \(M\).

Question 11 EITHER alternative.

A particle \(P\) of mass \(3 m\) is attached to one end of a light elastic spring of natural length \(a\) and modulus of elasticity \(k m g\). The other end of the spring is attached to a fixed point \(O\) on a smooth plane that is inclined to the horizontal at an angle \(\alpha\), where \(\sin \alpha=\frac{2}{3}\). The system rests in equilibrium with \(P\) on the plane at the point \(E\). The length of the spring in this position is \(\frac{5}{4} a\).
(i) Find the value of \(k\).

The particle \(P\) is now replaced by a particle \(Q\) of mass \(2 m\) and \(Q\) is released from rest at the point \(E\).
(ii) Show that, in the resulting motion, \(Q\) performs simple harmonic motion. State the centre and the period of the motion.

(iii) Find the least tension in the spring and the maximum acceleration of \(Q\) during the motion.

\(6 \quad A\) and \(B\) are two fixed points at a distance \(22 a\) apart, with \(B\) vertically below \(A\). A light elastic string of natural length \(4 a\) and modulus of elasticity \(5 m g\) has one end attached to \(A\) and the other end attached to a particle \(P\) of mass km . Another light elastic string of natural length \(8 a\) and modulus of elasticity \(6 m g\) has one end attached to \(B\) and the other end attached to \(P\). Particle \(P\) is vertically above \(B\). (a) Show that, when the system is in equilibrium, \(B P=\frac{57 a-2 a k}{4}\).

The particle \(P\) is pulled vertically upwards so that \(B P=18 a\), and is then released from rest. In its subsequent motion, \(P\) first comes to instantaneous rest at the point where \(B P=8 a\). (b) Find the value of \(k\).

A particle of weight 10 N is attached to one end of a light elastic string. The other end of the string is attached to a fixed point \(A\) on a horizontal ceiling. A horizontal force of 7.5 N acts on the particle. In the equilibrium position, the string makes an angle \(\theta\) with the ceiling (see diagram). The string has natural length 0.8 m and modulus of elasticity 50 N .
(a) Find the tension in the string.

(b) Find the vertical distance between the particle and the ceiling.