A2 October 2021 Paper 1 Q11

OCR ACurrent spec5 marksSecond Order Differentials

11 The displacement of a door from its equilibrium (closed) position is measured by the angle, \(\theta\) radians, which the door makes with its closed position. The door can swing either side of the equilibrium position so that \(\theta\) can take positive and negative values. The door is released from rest from an open position at time \(t = 0\).

A proposed differential equation to model the motion of the door for \(t \geqslant 0\) is

\(\dfrac{\mathrm{d}^2\theta}{\mathrm{d}t^2} + \lambda\dfrac{\mathrm{d}\theta}{\mathrm{d}t} + 3\theta = 0\) where \(\lambda\) is a constant and \(\lambda \geqslant 0\).

(a)

(i) According to the model, for what value of \(\lambda\) will the motion of the door be simple harmonic? [1]

(ii) Explain briefly why modelling the motion of the door as simple harmonic is unlikely to be realistic. [1]

(b) Find the range of values of \(\lambda\) for which the model predicts that the door will never pass through the equilibrium position. [2]

(c) Sketch a possible graph of \(\theta\) against \(t\) when \(\lambda\) lies outside the range found in part (b) but the motion is not simple harmonic. [1]

Scheme	Marks	AO
(i) For SHM \(\lambda = 0\)	B1	3.3
	[1]
(ii) The door should close, but in SHM the motion continues indefinitely	B1	3.5b
	[1]

Mark scheme (b)

Scheme	Marks	AO
Over- or critical- damping implies \(\lambda^2 - 12 \geqslant 0\) So \(\lambda \geqslant 2\sqrt{3}\)	M1 A1	3.3 3.4
	[2]

Notes

M1: Consider discriminant with \(\geqslant\) or \(\gt\)

Ignore \(\lambda \leqslant -2\sqrt{3}\)

Graph of theta against t: starts at a positive value on the theta-axis with zero gradient, falls through the t-axis to a minimum below it, rises to a smaller peak above the axis, and then levels off towards the t-axis — Mark scheme (c)