Answer to Question #157509 in Trigonometry for Carty

Question

A uniform beam AB of length a and weight W is free to turn in a vertical plane about a hinge at A and is supported in a horizontal position by a string attached to the beam at a point D at a distance a/2 from A and to a point F at a height b vertically above A. Show that the tension, in thd string is [w*sqrt(a^2 + 9b^2)] / 2b.
Find in terms of w, a and b, the magnitude of the reaction at the hinge. Find also the tangent this reaction makes with the horizontal.

Answer 1

Since the system is in equilibrium,

$Tsin\theta × l = W × \frac{a}{2}\\ but,\\ l = \sqrt{(\frac{a}2)^2 + b^2} = \sqrt{\frac{a^2}4 + b^2}$

$T × \sqrt{\frac{a^2}4 + b^2}× sin\theta = \frac{W×a}{2}$

$T = \dfrac{Wa}{2 \sqrt{\frac{a^2}4 + b^2} × sin\theta}$

Cross multiplying by the root numbers,

$T = \dfrac{Wa}{2 \sqrt{\frac{a^2}4 + b^2} × sin\theta} × \dfrac{ \sqrt{\frac{a^2}4 + b^2}}{ \sqrt{\frac{a^2}4 + b^2}}$

$T = \dfrac{Wa × \sqrt{\frac{a^2}4 + b^2}}{2sin \theta \ (\frac{a^2}4 + b^2)}$

Since the angle of inclination was not given, the question cannot be ccompleted.