Edexcel A-Level Mathematics A2 M2.2 Centres of Mass Questions

M( x axis)

M1

Need all terms. Dimensionally consistent. Condone if m missing throughout.

Accept as part of a vector equation

$$\begin{aligned} 2 m \times-+\times+\times=\times(2) 3 m 24 m 3 k \quad 9 m \quad x x=\frac{2+12 k}{9} * \end{aligned}$$

A1*

Obtain given result

2

M(y axis)
Need all terms. Dimensionally consistent. Might be seen as part of a vector equation in (a). It does not score any marks until referred to in part (b). Condone if m missing throughout.

$$\begin{aligned} 2 m \times+\times-+\times=\times 53 m(3) 4 m k \quad 9 m \quad y \square-14+k \square \square y=\square \square \end{aligned}$$
Correct unsimplified equation. Allow if m missing
throughout.

Form and solve equation in k
$$(2+12 k++=2 \quad 8 k \quad 27)$$
Use their y ānd $x \overline{+}^{-}=2 y^{-} \quad 3$

Dependent on the two preceding M marks

$k=\frac{23}{20}(1.15)$
Correct answer only

4

(6)

Area ratios and distances from the axis through X:
Large disc: area $\pi R^2$, distance 0.
Small disc: area $\pi r^2$, distance R-r.
Template: area $\pi R^2-\pi r^2$, distance $\pm kr$.
Correct area ratios and distances seen or implied. Allow +kr or -kr.
Moments about the axis through X.
Or moments about a parallel axis. Need all terms but condone sign errors. Dimensionally consistent.
$$0\times\pi R^2-\pi r^2(R-r)=(\pi R^2-\pi r^2)(-kr)$$
Correct unsimplified equation.
$$r=\frac{k}{1-k}R$$
A1*

Use the correct inequality.
0<r<R
Using $r=\frac{k}{1-k}R$:
0<k<1-k$$0<k<\frac12$$
Correct only. Only the right-hand value is needed;
A0 with an incorrect left-hand value.

$$k=\frac49\Rightarrow r=\frac45R$$$$\tan\alpha=\frac{R}{kr} =\frac{R}{\frac49\times\frac45R} =\frac{45}{16}$$$$\alpha=70.4^\circ\ \text{(awrt)}$$