| Exam Board | OCR |
|---|---|
| Module | Further Pure Core 1 (Further Pure Core 1) |
| Year | 2018 |
| Session | March |
| Marks | 6 |
| Topic | Second order differential equations |
| Type | Standard non-homogeneous with trigonometric RHS |
| Difficulty | Moderate -0.5 This is a standard second-order linear homogeneous differential equation with constant coefficients (simple harmonic motion). Part (i) requires recognizing the characteristic equation gives λ²=-4, leading to sinusoidal solutions. Part (ii) applies initial conditions θ(0)=θ₀ and θ'(0)=0, which is routine. Part (iii) asks for model limitations (small angle assumption), which is straightforward recall. While this is Further Maths content, it's a textbook SHM problem requiring no novel insight—slightly easier than average A-level difficulty overall. |
| Spec | 4.10d Second order homogeneous: auxiliary equation method4.10f Simple harmonic motion: x'' = -omega^2 x |
6 One end of a light inextensible string is attached to a small mass. The other end is attached to a fixed point $O$. Initially the mass hangs at rest vertically below $O$. The mass is then pulled to one side with the string taut and released from rest. $\theta$ is the angle, in radians, that the string makes with the vertical through $O$ at time $t$ seconds and $\theta$ may be assumed to be small.
The subsequent motion of the mass can be modelled by the differential equation
$$\frac { \mathrm { d } ^ { 2 } \theta } { \mathrm {~d} t ^ { 2 } } = - 4 \theta$$
(i) Write down the general solution to this differential equation.\\
(ii) Initially the pendulum is released from rest at an angle of $\theta _ { 0 }$. Find the particular solution to the equation in this case.\\
(iii) State any limitations on the model.
\hfill \mbox{\textit{OCR Further Pure Core 1 2018 Q6 [6]}}