Question

The drawing shows a uniform horizontal beam attached to a vertical wall by a frictionless hinge and supported from below at an angle θ = 43o by a brace that is attached to a pin. The beam has a weight of 336 N. Three additional forces keep the beam in equilibrium. The brace applies a force to the right end of the beam that is directed upward at the angle θ with respect to the horizontal. The hinge applies a force to the left end of the beam that has a horizontal component and a vertical component . Find the magnitudes of these three forces.

Answer 1

Answer:

$F_b = 246.3 N$

$F_x = 180.15 N$

$F_y = 168 N$

Explanation:

Let the force exerted by the brace is given as

$F_b$ along its direction of length

also the two components at the end of the pin is given as

$F_x , F_y$

now by vertical force balance we have

$F_y + F_b sin43 = mg$

$Fx = Fb cos43$

Now by torque balance about the hinge point we have

$F_bsin43 \times L = mg \times\frac{L}{2}$

so we have

$F_b = \frac{mg}{2 sin43}$

$F_b = \frac{336}{2sin43}$

$F_b = 246.3 N$

now we have

$F_x = 246.3 cos43$

$F_x = 180.15 N$

similarly by Y direction equation

$F_y + 246.3 sin43 = 336$

$F_y = 168 N$