As shown in the diagram, particles $$\(A\)$$ and $$\(B\)$$ of masses $$\(2 \mathrm{~kg}\)$$ and $$\(3 \mathrm{~kg}\)$$ respectively are attached to the ends of a light inextensible string. The string passes over a small fixed smooth pulley which is attached to the top of two inclined planes. Particle $$\(A\)$$ is on plane $$\(P\)$$, which is inclined at an angle of $$\(10^{\circ}\)$$ to the horizontal. Particle $$\(B\)$$ is on plane $$\(Q\)$$, which is inclined at an angle of $$\(20^{\circ}\)$$ to the horizontal. The string is taut, and the two parts of the string are parallel to lines of greatest slope of their respective planes. It is given that plane $$\(P\)$$ is smooth, plane $$\(Q\)$$ is rough, and the particles are in limiting equilibrium. Find the coefficient of friction between particle $$\(B\)$$ and plane $$\(Q\)$$. ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................

Mathematics
IGCSE&ALevel
CAIE
Exam No:9709_w20_qp_43 Year:2020 Question No:7(a)

Answer:

\( \)  \( [T=2 g \sin 10]$ or $[3 g \sin 20=F+T] \)
\(T=2 g \sin 10\) and \(3 g \sin 20=F+T\)
\(R=30 \cos 20(=28.19 \ldots)\)
\(\mu=\frac{3 g \sin 20-2 g \sin 10}{30 \cos 20}\)
\(\mu=0.241(=0.2407 \ldots)\)

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.)
4.1.4 understand that a contact force between two surfaces can be represented by two components, the normal component and the frictional component
4.1.6 understand the concepts of limiting friction and limiting equilibrium, recall the definition of coefficient of friction, and use the relationship F = nR or F G nR, as appropriate

Solution:

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