Question #157509
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.
1
Expert's answer
2021-02-09T01:11:48-0500




Since the system is in equilibrium,

Tsinθ×l=W×a2but,l=(a2)2+b2=a24+b2Tsin\theta × l = W × \frac{a}{2}\\ but,\\ l = \sqrt{(\frac{a}2)^2 + b^2} = \sqrt{\frac{a^2}4 + b^2}


T×a24+b2×sinθ=W×a2T × \sqrt{\frac{a^2}4 + b^2}× sin\theta = \frac{W×a}{2}


T=Wa2a24+b2×sinθT = \dfrac{Wa}{2 \sqrt{\frac{a^2}4 + b^2} × sin\theta}


Cross multiplying by the root numbers,


T=Wa2a24+b2×sinθ×a24+b2a24+b2T = \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=Wa×a24+b22sinθ (a24+b2)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.


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