Hooke's law：練習問題

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.

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 \(l\).

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 \( u \).