Particle at midpoint of string between two horizontal fixed points: horizontal surface motion

3 \includegraphics[max width=\textwidth, alt={}, center]{18398d27-15eb-4515-8210-4f0f614d5b28-2_247_839_1375_653} A light elastic string of natural length 1.2 m and modulus of elasticity 24 N is attached to fixed points \(A\) and \(B\) on a smooth horizontal surface, where \(A B = 1.2 \mathrm {~m}\). A particle \(P\) is attached to the mid-point of the string. \(P\) is projected with speed \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) along the surface in a direction perpendicular to \(A B\) (see diagram). \(P\) comes to instantaneous rest at a distance 0.25 m from \(A B\).

1. Show that the mass of \(P\) is 0.8 kg .
2. Calculate the greatest deceleration of \(P\).

3. A light elastic string has natural length \(2 l\) and modulus of elasticity \(4 m g\). One end of the string is attached to a fixed point \(A\) and the other end to a fixed point \(B\), where \(A\) and \(B\) lie on a smooth horizontal table, with \(A B = 4 l\). A particle \(P\) of mass \(m\) is attached to the mid-point of the string. The particle is released from rest at the point of the line \(A B\) which is \(\frac { 5 l } { 3 }\) from \(B\). The speed of \(P\) at the mid-point of \(A B\) is \(V\).

1. Find \(V\) in terms of \(g\) and \(L\).
2. Explain why \(V\) is the maximum speed of \(P\).
   (Total 9 marks)

14 \includegraphics[max width=\textwidth, alt={}, center]{22640c3b-792f-4003-a4f8-78220efd73b0-5_86_1589_1297_278} A particle of mass 0.05 kg is attached to two identical light elastic strings, each of natural length 1.2 m and modulus of elasticity 0.6 N . The other ends of the strings are attached to points \(A\) and \(E\) on a smooth horizontal table. The distance \(A E\) is 2 m and points \(B , C\) and \(D\) lie between \(A\) and \(E\) so that \(A B = 0.7 \mathrm {~m} , B C = 0.1 \mathrm {~m} , C D = 0.4 \mathrm {~m}\) and \(D E = 0.8 \mathrm {~m}\) (see diagram). Initially the particle is held at \(B\) and it is then released. In the subsequent motion the displacement of the particle from \(C\), in the direction of \(A\), is denoted by \(x \mathrm {~m}\).

1. Find the equation of motion for the particle when it is between \(B\) and \(C\).
2. Find the velocity of the particle when it is at \(C\).
3. Find the total time that elapses before the particle first returns to \(B\).

\includegraphics{figure_4} A light elastic string has natural length \(8a\) and modulus of elasticity \(5mg\). A particle \(P\) of mass \(m\) is attached to the midpoint of the string. The ends of the string are attached to points \(A\) and \(B\) which are a distance \(12a\) apart on a smooth horizontal table. The particle \(P\) is held on the table so that \(AP = BP = L\) (see diagram). The particle \(P\) is released from rest. When \(P\) is at the midpoint of \(AB\) it has speed \(\sqrt{80ag}\).

1. Find \(L\) in terms of \(a\). [5]
2. Find the initial acceleration of \(P\) in terms of \(g\). [3]

\includegraphics{figure_3} A light elastic string has natural length \(8a\) and modulus of elasticity \(5mg\). A particle \(P\) of mass \(m\) is attached to the midpoint of the string. The ends of the string are attached to points \(A\) and \(B\) which are a distance \(12a\) apart on a smooth horizontal table. The particle \(P\) is held on the table so that \(AP = BP = L\) (see diagram). The particle \(P\) is released from rest. When \(P\) is at the midpoint of \(AB\) it has speed \(\sqrt{80ag}\).

1. Find \(L\) in terms of \(a\). [5]
2. Find the initial acceleration of \(P\) in terms of \(g\). [3]