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A uniform plank PQ of length 20m weighing 55N on a ceiling with an inextensible wire
3m away from P. To keep the plank horizontally, it is made to rest on a support 8m from Q.
Calculate the tension T on the wire and the weight W exerted on the support.
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Let the tension be T newtons, and let W be the support force at the support point.
Then we have this equilibrium equation
T + W = 55 newtons. (1)
Second equation is equality of rotation moments around the center of the plank.
The leg of the force T is 20/2-3 = 7 meters; the leg of the force W is 20/2-8 = 2 meters.
The rotation moment of the force T is 7*T N*m. The rotation moment of the force W is 2*W N*m.
The equation for rotation moments is
7T = 2W. (2)
So, we have two equations, (1) and (2), for two unknowns, T and W.
To solve, express T = 55-W from (1) and substitute it in equation (2). You will get
7*(55-W) = 2W,
385 - 7W = 2W
385 = 2W + 7W
385 = 9W
W = 385/9 = 42.778 Newtons (rounded).
Hence, force T is T = 55-42.778 = 12.222 Newtons.
ANSWER. The tension is 12.222 N. The support force is 42.778 N.
Solved.
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Important post-solution notice
Considering rotation moments around the central point of the plank,
we remove the influence and the contribution of the own weight of the plank
to rotation moments. It is where the condition is used that the plank is uniform.