Cambridge International AS Level · Mathematics - Further (9231)
5 multiple-choice questions marked as you go, and 3 written questions with worked solutions. All on Hooke's law.
A particle of mass \(m\) is attached to one end of a light elastic string of natural length \(l\) and modulus of elasticity \(\lambda = 2mg\). The other end of the string is fixed to a point \(O\) on a smooth horizontal table. The particle is held at a point such that the string is slack, and is then projected away from \(O\) with speed \(u = \sqrt{3gl}\). Find the maximum extension of the string during the motion.
A particle of mass \(m\) is attached to one end of a light elastic string of natural length \(l\) and modulus \(mg\). The other end is fixed to a point \(O\). The particle is projected vertically downwards from \(O\) with speed \(\sqrt{2gl}\). Find the maximum extension reached by the particle.
A particle of mass \(m\) is attached to one end of a light elastic string of natural length \(L\) and modulus of elasticity \(\lambda\). The other end of the string is attached to a fixed point \(O\). The particle is released from rest at \(O\). Given that the particle first comes to instantaneous rest after falling a total distance of \(3L\), find the value of \(\lambda\) in terms of \(m\) and \(g\).
A particle of mass \(m\) is attached to one end of a light elastic string of natural length \(l\) and modulus of elasticity \(4mg\). The other end of the string is attached to a fixed point \(A\) on a smooth horizontal table. The particle is projected from \(A\) with speed \(u\). Find the minimum value of \(u\) such that the string reaches a total length of \(1.5l\).
A particle of mass \(m\) is attached to one end of a light elastic string of natural length \(l\) and modulus \(mg\). The other end of the string is attached to a fixed point \(O\) on a smooth plane inclined at \(30^\circ\) to the horizontal. The particle is released from rest at \(O\) and slides down the line of greatest slope. Find the maximum extension of the string.
A particle of mass \(m\) is attached to one end of a light elastic string of natural length \(L\) and modulus of elasticity \(5mg\). The other end of the string is fixed to a point \(O\). The particle is released from rest at a point \(L\) vertically below \(O\). Find the maximum extension of the string during the subsequent motion.
Write your answer out first, then check it against the worked solution.
A light elastic string of natural length \(l\) and modulus of elasticity \(3mg\) is attached at one end to a fixed point \(O\). A particle of mass \(m\) is attached to the other end and hangs in equilibrium. Find the extension of the string in terms of \(l\).
Write your answer out first, then check it against the worked solution.
A particle of mass \( m \) is attached to one end of a light elastic string of natural length \( L \) and modulus of elasticity \( 2mg \). The other end of the string is attached to a fixed point \( O \) on a smooth horizontal table. The particle is held at a point \( A \) such that \( OA = L \) and is then projected away from \( O \) with speed \( u \).
(a) Find the extension of the string when the particle first comes to instantaneous rest in terms of \( u, L, \) and \( g \).
(b) Given that the maximum speed reached by the particle in its subsequent motion is \( \sqrt{gL} \), find the value of \( u \).
Write your answer out first, then check it against the worked solution.
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