A purple beam is hinged to a wall to hold up a blue sign. The beam has a mass of

Question

A purple beam is hinged to a wall to hold up a blue sign. The beam has a mass of mb = 6 kg and the sign has a mass of ms = 16.3 kg. The length of the beam is L = 2.81 m. The sign is attached at the very end of the beam, but the horizontal wire holding up the beam is attached 2/3 of the way to the end of the beam. The angle the wire makes with the beam is = 30.9°.

1) What is the tension in the wire?   

In Newtons

2 )What is the net force the hinge exerts on the beam?

In Newtons

3) The maximum tension the wire can have without breaking is T = 954 N.

What is the maximum mass sign that can be hung from the beam?

In Kg

4) What else could be done in order to be able to hold a heavier sign?

-while still keeping it horizontal, attach the wire to the end of the beam

-keeping the wire attached at the same location on the beam, make the wire perpendicular to the beam

-attach the sign on the beam closer to the wall

-shorten the length of the wire attaching the sign to the beam

Explanation / Answer

Horizontal distance of 6.0 kg from the hinge is 1.405 cos 30.9 = 1.206 m
Horizontal distance of 16.3 kg from the hinge is 2.81 cos 30.9 = 2.412 m
The vertical distance of the tension T from the hinge is (2/3)*2.81*sin 30.9 = 0.962 m

1.
Taking moment about the hinge,
0.962*T = (6*1.206 + 16.3*2.412)*9.8
T =474.23 N
-----------------------------
2
sqrt( 474.23^2 +(6*9.8)^2 +( 16.3*9.8)^2 )
503.85 N at an angle 30.9°. Above the horizontal.
--------------------------------------...
3.
0.962*954 = (6*1.206 + m *2.412)*9.8

m = 35.83 kg.

4.

attach the sign on the beam closer to the wall, --> good
keeping the wire attached at the same location on the beam, make the wire perpendicular to the beam --> also good

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