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The diagram shows three particles $A$, $B$ and $C$ hanging freely in equilibrium, each being attached to the end of a string. The other ends of the three strings are tied together and are at the point $X$. The strings carrying $A$ and $C$ pass over smooth fixed horizontal pegs ${P_1}$ and ${P_2}$ respectively. The weights of $A$, $B$ and $C$ are $5.5{\text{ }}N$, $7.3{\text{ }}N$ and $W{\text{ }}N$ respectively, and the angle respectively, and the angle ${P_1}X{P_2}$ is a right angle. Find the angle $A{P_1}X$ and the value of $W$.

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For correct $\Delta $ or resolve $X{P_1}$

and $\cos \alpha  = 5.5/7.3$;

or $5.5/\sin \left( {{{90}^ \circ } + \alpha } \right) = 7.3/\sin {90^ \circ }$ (Lami);

or $5.5\cos \alpha  + W\sin \alpha  = 7.3$

and $5.5\sin \alpha  = W\cos \alpha $.

Angle $A{P_1}X = {41.1^ \circ }$ or ${0.718^ \circ }$

For correct triangle and ${W^2} = {7.3^2} - {5.5^2}$;

or $W/\sin \left( {{{180}^ \circ } - {{41.1}^ \circ }} \right) = 7.3/\sin {90^ \circ }$;

or $W\sin {41.1^ \circ } = 7.3 - 5.5\cos {41.1^ \circ }$

or $W\cos {41.1^ \circ } = 5.5\sin {41.1^ \circ }$

$W = 4.8$

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