CAIE A-Level Mathematics 4.4.4 Connected-Particle Models

Practise modelling particles joined by a light inextensible string, using common acceleration and separate or whole-system force equations to determine motion and tension.

Exam points

draw a separate force model for each particle and assign one consistent acceleration direction
use equal acceleration magnitudes while the light inextensible string remains taut
solve a whole-system equation for acceleration, then substitute into one body equation for tension

[Maximum number: 5]

Figure for Question 4.4.4—Connected-particle models question 1 — CAIE A-Level Mathematics A2

Two particles A and B, of masses $3 m \mathrm{~kg}$ and $2 m \mathrm{~kg}$ respectively, are attached to the ends of a light inextensible string. The string passes over a fixed smooth pulley which is attached to the edge of a plane. The plane is inclined at an angle $\theta$ to the horizontal. A lies on the plane and B hangs vertically, 0.8 m above the floor, which is horizontal. The string between A and the pulley is parallel to a line of greatest slope of the plane (see diagram). Initially A and B are at rest.

Show that the coefficient of friction between A and the plane is $\frac{1}{10} \sqrt{3}$ .

$R=3 m g \cos 30$
B1
Use of $F=\mu R$
M1
$2 m g-T=0.1 \times 2 m$ OR $T-3 m g \sin 30-\mu \times 3 m g \cos 30=0.1 \times 3 m$
M1
$2 m g-0.2 m-3 m g \sin 30-\mu \times 3 m g \cos 30=0.1 \times 3 m$
M1
$\mu=\frac{\sqrt{3}}{10}$
A1

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4.4.4—Connected-particle models question 1