Magnitude and sense of the force applied at A to reduce the reaction at B to zero.

Question

A light horizontal beam $AB$, of length $9m$, supported at its ends by a force $S$ acting vertically and a force $R$ acting at an angle α to the line of the beam. A force of $30N$ is applied to the beam, at an angle of $30^o$, $3m$ from $B$. The beam is in equilibrium and $S=10N$, $α=arctan(\frac{1}{3√3})$ and $R=10√7N$.

Calculate the magnitude and sense of the necessary moment that would have to be applied at $A$ to reduce the reaction at $B$ to zero.

(Answer given as $90Nm$ anticlockwise)

What is the best approach to the last part of the question? Summing the forces at each point or taking moments?

Answer 1

Assuming the conditions are the same as in the first part of the question, where $R$, $\alpha$ and $S$ had to be worked out, we now add an extra force at $A$ to create an anticlockwise moment, which is pointing directly downwards at $A$.

We then take moments about $B$, and with $Rsinα = 5N$:

$$M(B): -9Rsinα - 3*30sin30^o + 9F = 0$$

So $$-45 -45 = -9F$$

So $$ -90 = -9F$$

Therefore $$ 9F = 90 Nm$$

So the necessary moment about $A$ to reduce the reaction at $B$ to zero is $90Nm$ anticlockwise.

(It turns out that $F=10N$, which is the same as $S$, so $9F$ is the necessary moment, in $Nm$.)