A2 October 2021 Q1

EdexcelCurrent spec8 marksCentres of Mass

Figure 1: letter P: shaded rectangle OABDE with OA = a along the bottom and OE = 4a up the left side; a shaded semicircle BCD on the upper part of the right side AD, with diameter BD = 2a and AB = 2a — Figure 1

A letter P from a shop sign is modelled as a uniform plane lamina which consists of a rectangular lamina, \(OABDE\), joined to a semicircular lamina, \(BCD\), along its diameter \(BD\).

\(OA = ED = a\), \(AB = 2a\), \(OE = 4a\), and the diameter \(BD = 2a\), as shown in Figure 1.

Using the model,

(a) find, in terms of \(\pi\) and \(a\), the distance of the centre of mass of the letter P,
from

(i) \(OE\)

(ii) \(OA\) (6)

The letter P is freely suspended from \(O\) and hangs in equilibrium. The angle between \(OE\) and the downward vertical is \(\alpha\).

Using the model,

(b) find the exact value of \(\tan\alpha\) (2)

Mark scheme (a)

Scheme	Marks	AO
Mass ratios: \(4a^2,\ \ \dfrac{1}{2}\pi a^2,\ \ \left(4a^2 + \dfrac{1}{2}\pi a^2\right)\)	B1	1.2
\(x\): \(\dfrac{1}{2}a,\ \ a + \dfrac{4a}{3\pi},\ \ \bar{x}\) \(y\): \(2a,\ \ 3a,\ \ \bar{y}\)	B1	1.2
Moments about \(OE\)	M1	3.1b
\(\bar{x} = \dfrac{(16 + 3\pi)a}{3(8 + \pi)}\)	A1	1.1b
Moments about \(OA\)	M1	3.1b
\(\bar{y} = \dfrac{(16 + 3\pi)a}{(8 + \pi)}\)	A1	1.1b
	(6)

Notes

B1: All correct

B1: Distances could be measured from a parallel axis

M1: All terms needed and must be dimensionally correct

A1: cao (must be in terms of \(\pi\) and \(a\))

M1: All terms needed and must be dimensionally correct

A1: cao (must be in terms of \(\pi\) and \(a\))

Mark scheme (b)

Scheme	Marks	AO
\(\tan\alpha = \dfrac{\bar{x}}{\bar{y}}\) and substitute for their \(\bar{x}\) and \(\bar{y}\)	M1	3.1b
\(\tan\alpha = \dfrac{1}{3}\)	A1	1.1b
	(2)
	(8 marks)

Notes

M1: Do not allow the reciprocal

A1: cao