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, $$$0.8 \mathrm{~m}$$$ 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. Given that the plane is smooth, find the value of $$$\theta$$$ for which $$$A$$$ remains at rest. ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................

Mathematics

IGCSE&ALevel

CAIE

Exam No：9709_s20_qp_43 Year：2020 Question No：7(a)

Answer:

$
T-2 m g=0
$
$3 m g \sin \theta-T=0$
(M1 for resolving forces parallel to the plane and solving for $\theta$ )
$
\theta=41.8(41.810 \ldots)
$

Knowledge points:

4.1.1 identify the forces acting in a given situation； e.g. by drawing a force diagram.

4.1.2 understand the vector nature of force, and find and use components and resultants； Calculations are always required, not approximate solutions by scale drawing.

4.1.3 use the principle that, when a particle is in equilibrium, the vector sum of the forces acting is zero, or equivalently, that the sum of the components in any direction is zero （Solutions by resolving are usually expected, but equivalent methods (e.g. triangle of forces, Lami’s Theorem, where suitable) are also acceptable; these other methods are not required knowledge, and will not be referred to in questions.）

Answer:

Knowledge points:

Solution:

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