Further Math A2s 3.2.4 Centre of mass

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Figure for Question CAIE A-Level Further Math A2 3.2.4 Centre of Mass Questions question 1 — CAIE A-Level Further Math A2

A uniform lamina is in the form of an isosceles triangle ABC in which A C=2 a and angle $A B C=90^{\circ}$ . The point D on AB is such that the ratio D B: A B=1: k. The point E on CB is such that DE is parallel to AC. The triangle DBE is removed from the lamina (see diagram).

(a)

Find, in terms of k, the distance of the centre of mass of the remaining lamina ADEC from the midpoint of AC.

When the lamina ADEC is freely suspended from the vertex A, the edge AC makes an angle $\theta$ with the downward vertical, where $\tan \theta=\frac{5}{18}$ .

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Mass is proportional to area

Figure for Answer to Question (a) — CAIE A-Level Further Math A2

At least two areas correct, at least one distance correct

$\bar{x}\left(-\frac{1}{2} \times 2\left(\frac{a}{k}\right)^{2}+a^{2}\right)=\frac{1}{3} a \times a^{2}-\left(a-\frac{2}{3}\left(\frac{a}{k}\right)\right) \times\left(\frac{a}{k}\right)^{2}$
Moments equation, dimensionally correct, correct number
of terms.

$\bar{x}=\frac{a\left(k^{2}+k-2\right)}{3 k(k+1)}$
Allow unsimplified single fraction $\frac{a\left(k^{3}-3 k+2\right)}{3 k\left(k^{2}-1\right)}$ .

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(b)

Find the value of k.

Figure for Question (b) — CAIE A-Level Further Math A2

Two smooth vertical walls meet at right angles. The smooth sphere A, with mass m, is at rest on a smooth horizontal surface and is at a distance d from each wall. An identical smooth sphere B is moving on the horizontal surface with speed u at an angle $\theta$ with the line of centres when the spheres collide (see diagram). After the collision, the spheres take the same time to reach a wall. The coefficient of restitution between the spheres is $\frac{1}{2}$ .

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$\tan \theta=\frac{\bar{x}}{a} \quad\left[=\frac{5}{18}\right]$
With their $\bar{x}$ from part (a).

So $18\left(k^{2}+k-2\right)=5\left(3 k^{2}+3 k\right)$ $k^{2}+k-12=0$
Obtain a polynomial in k only, e.g. $k^{3}-13 k+12=0$ ,
may be implied.

(k+4)(k-3)=0,

k=3 only
CWO

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CAIE A-Level Further Math A2 3.2.4 Centre of Mass Questions question 1

Question (a)

Question (b)