M5 June 2017 Q4

4. A uniform lamina \(PQR\) of mass \(m\) is in the shape of an isosceles triangle, with \(PQ = PR = 5a\) and \(QR = 6a\). The midpoint of \(QR\) is \(T\).

(a) Show, using integration, that the moment of inertia of the lamina about an axis which passes through \(P\) and is parallel to \(QR\), is \(8ma^2\). (6)

(b) Show, using integration, that the moment of inertia of the lamina about an axis which passes through \(P\) and \(T\), is \(1.5ma^2\).

[You may assume without proof that the moment of inertia of a uniform rod, of mass \(m\) and length \(2l\), about an axis perpendicular to the rod through its midpoint is \(\dfrac{1}{3}ml^2\)]

(4)

The lamina is now free to rotate in a vertical plane about a fixed smooth horizontal axis \(A\) which passes through \(P\) and is perpendicular to the plane of the lamina. The lamina makes small oscillations about its position of stable equilibrium.

(c) By writing down an equation of rotational motion for the lamina as it rotates about \(A\), find the approximate period of these small oscillations. (6)

Mark scheme (a)

Scheme	Marks
\(\delta A = \dfrac{3x}{2}\delta x\)	M1 A1
\(\delta m = \dfrac{3x}{2}\delta x\dfrac{m}{12a^2} = \dfrac{mx\,\delta x}{8a^2}\)	M1
\(\delta I = \dfrac{mx\,\delta x}{8a^2}x^2 = \dfrac{mx^3\,\delta x}{8a^2}\)	M1
\(I_y = \displaystyle\int_0^{4a} \dfrac{mx^3\,\mathrm{d}x}{8a^2} = 8ma^2\) PRINTED ANSWER	DM1 A1
	(6)

Notes

First M1 for attempt at area of strip in terms of a variable and an increment

First A1 if it’s correct

Second M1 for multiplying their area by correct mass per unit area

Third M1 for \(\delta mx^2\) or other appropriate \(\delta I\)

Fourth DM1, dependent on 3^rd M1 for integrating their \(\delta I\) between appropriate limits

Second A1 for PRINTED ANSWER

Mark scheme (b)

Scheme	Marks
\(\delta I = \dfrac{1}{3}\dfrac{mx\,\delta x}{8a^2}\left(\dfrac{3x}{4}\right)^2 = \dfrac{3mx^3\,\delta x}{128a^2}\)	M1 A1
\(I_y = \displaystyle\int_0^{4a} \dfrac{3mx^3\,\mathrm{d}x}{128a^2} = 1.5ma^2\) PRINTED ANSWER	DM1 A1
	(4)

Notes

First M1 for using appropriate expression for \(\delta I\)

First A1 for a correct expression

Second DM1, dependent on 1st M, for integrating their \(\delta I\) between appropriate limits

Second A1 for PRINTED ANSWER

Mark scheme (c)

Scheme	Marks
Perp axes: \(I = 1.5ma^2 + 8ma^2 = 9.5ma^2\)	M1 A1
\(mg\dfrac{8a}{3}\sin\theta = -\dfrac{19ma^2}{2}\ddot{\theta}\)	M1 A1
For small \(\theta\), \(\ \ddot{\theta} = -\dfrac{16g}{57a}\theta\)	M1
\(T = 2\pi\sqrt{\dfrac{57a}{16g}}\)	A1
	(6)
	(16 marks)

Notes

First M1 for use of perp axes

First A1 for a correct MI

Second M1 for moments equation with usual rules

Second A1 for a correct equation

Third M1 for use of small angle approximation to obtain an SHM equation

Third A1 for answer.