centre of gravity of non uniform beam

a)How can i determine the centre of gravity of a non uniform beam?
b)If a non uniform plank is supported by two force meters A and B in which A reads more force than B,where will be the centre of gravity?Why and how?Please help me through this.

Reply 1

Drummy

a) to find the cog of a non uniform beam just find where it balances on a knife edge
b) if a beam is supported by two force meters and one reads more than the other then it is certainly a non uniform beam. To solve this

i) draw a diagram of the beam and add vertical arrows for

F_A, F_B and W

where W is the weight of the beam. Draw this closer to A (if A reads more than B) label the distance fro A to W, x. The length of the beam is l.
ii) take moments about point A
this will give

Wx = F_B l

therefore

x=\frac{F_B l}{W} =\frac{F_B l}{F_A + F_B}

if we know what the force meters read and the length of the beam then we have found x, the distance of the cog from A.
further fiddling reveals

(1+\frac{F_A}{F_B}) = \frac{l}{x}

Reply 2

roshanhero

Thanks but the thing I don't understand is why we draw W nearer to A.Is it because it reads more force.

Reply 3

Drummy

Imagine two people are carrying something like a piano. Someone is going to get the "heavy end" so they need to use a bigger force. It is the heavy end because the COG is closer to that end than the other, hence we draw the weight arrow closer to that end.