| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/2 (Pre-U Further Mathematics Paper 2) |
| Year | 2013 |
| Session | November |
| Topic | Simple Harmonic Motion |
| Type | String becomes slack timing |
| Difficulty | Challenging +1.2 This is a standard SHM problem with elastic strings requiring equilibrium analysis, deriving the SHM equation from Hooke's law, finding when tension becomes zero, and analyzing motion under gravity alone. While it involves multiple parts and careful tracking of reference points, each step follows well-established procedures taught in Further Maths mechanics with no novel insights required. |
| Spec | 4.10f Simple harmonic motion: x'' = -omega^2 x6.02g Hooke's law: T = k*x or T = lambda*x/l6.02h Elastic PE: 1/2 k x^26.02i Conservation of energy: mechanical energy principle6.05f Vertical circle: motion including free fall |
9 A light string, of natural length 0.5 m and modulus of elasticity 4 N , has one end attached to the ceiling of a room. A particle of mass 0.2 kg is attached to the free end of the string and hangs in equilibrium.\\
(i) Find the extension of the string when the particle is in the equilibrium position.
The particle is pulled down a further 0.5 m from the equilibrium position and released from rest. At time $t$ seconds the displacement of the particle from the equilibrium position is $x \mathrm {~m}$.\\
(ii) Show that, while the string is taut, the equation of motion is $\ddot { x } = - 40 x$.\\
(iii) Find the time taken for the string to become slack for the first time.\\
(iv) Show that the particle comes to instantaneous rest 0.125 m below the ceiling.
\hfill \mbox{\textit{Pre-U Pre-U 9795/2 2013 Q9}}