A2 June 2025 Q2

EdexcelCurrent spec7 marksCentres of Mass

Figure 2: curve y = 1/x with the region between x = 2 and x = 4 under the curve shaded — Figure 2

The shaded region shown in Figure 2 is bounded by the curve with equation \(y = \dfrac{1}{x}\), the line with equation \(x = 2\), the line with equation \(x = 4\), and the \(x\)-axis.
This region is rotated through \(360^\circ\) about the \(x\)-axis to form a solid.

This solid is used to model a uniform solid pedestal of height 2 m, upper radius 0.5 m and base radius 0.25 m.

Given that the volume of the pedestal is \(\dfrac{\pi}{4}\ \text{m}^3\)

(a) show that the centre of mass of the pedestal is \((4 - 4\ln 2)\) m from its base. (4)

Figure 3: the pedestal standing with its smaller plane base on a rough plane inclined at angle alpha degrees to the horizontal — Figure 3

Diagram not drawn to scale

The pedestal is placed on a rough plane that is inclined at an angle \(\alpha^\circ\) to the horizontal. The plane base of the pedestal is in contact with the inclined plane, as shown in Figure 3.
The inclined plane is sufficiently rough to prevent the pedestal from sliding.

Given that the pedestal is on the point of toppling,

(b) find the value of \(\alpha\) (3)

Mark scheme (a)

Scheme	Marks	AO
Moments about \(O\): \(\displaystyle\int_2^4 (\pi\rho)\,y^2 x\,\mathrm{d}x\)	M1	2.1
\(= (\pi\rho)\displaystyle\int_2^4 \dfrac{1}{x}\,\mathrm{d}x = \Big[(\pi\rho)\ln x\Big]_2^4 \quad \big(= (\pi\rho)\ln 2\big)\)	A1	1.1b
Complete strategy to find distance:	M1	3.1a
\(\dfrac{\pi}{4}d = \pi\ln 2 \quad \Rightarrow d = 4\ln 2\) (distance from base \(=\)) \(4 - 4\ln 2\) *	A1*	1.1b
	(4)

Notes

M1: Moments equation to obtain integral of the correct form (with or without limits; condone missing \(\mathrm{d}x\)).
Must be integrating \(y^2x\) and not just \(y\). Allow missing volume / \(\rho\) / \(\pi\)

A1: Correct unsimplified integration with correct limits seen (need not be substituted).
Allow missing volume / \(\rho\) / \(\pi\)

M1: Complete strategy to find a relevant distance (\(d\) or \(4 - d\)): use of moments equation, correct use of limits, division by volume. Condone calculation of volume.

A1*: Obtain given answer with correct working seen (condone poor integration notation).
\(\rho\) / \(\pi\) if seen must be used consistently and correctly.

Diagram: pedestal on the inclined plane on the point of toppling, with the vertical through the centre of mass passing through the lowest point of the base; distances 4 − 4ln2 and 0.25 and the angle alpha marked — Mark scheme (b)

Scheme	Marks	AO

Use of trigonometry to find a relevant angle	M1	3.1b
\(\tan\alpha^\circ = \dfrac{0.25}{4 - 4\ln 2}\)	A1	1.1b
\((\alpha =)\ 12\) or better	A1	1.1b
	(3)
	(7 marks)