PHYSICS helpA metal wire of diameter D stretches by 0.100mm when supporting a weight W. if the same length wire is used to support a weight W. if...

We know that the Young's modulus E, the coefficient of
linear stretch of the material of a wire is given by:

E =
(F*L0)/{A( L1-Lo)}, where F is the force tensile force , A is the cross sectional are of
the wire and L0 is the length of the wire beore applying the tensile stretch and L1  is
the length of the wire after applying the tensile force
F.

We know that Young's Modulus  E is fairly constant  for
a considerable variation of force.

Therefore ,
for

E =  FLo/{kd1^2*(L1-L0)}... (1) , as fross sectional
area is proportional to diameter, and d1 is the diameter of the wire for which we get
L1-L0 = 0.100mm.

E = 3FL0/{kd2^2 *(L2-L0)}....(2), where d2
is the diameter when applied a force of 3 times F. But L2 -L0 =
0.100mm.

Therefore , we cal equate the righ sides of the
equations (1) and (2), and L1-L0 = L2=L0 = 0.100mm = 10^(-4)
m

Therefore
,

FL0/{kd1^2*(L1-Lo)} =
3FL0/{kd2^2*(L2-L0)}

FL0/{kd1^2 *10^(-4)} =
3FL0/{kd2^2*10^(-4)}

After cross multiplication and
cancelltion of common factors , we get:

(d2)^2 =
3(d1)^2.

Therefore d2 = (sqrt3)
d1.

So the diameter of the wire should be sqrt 3 times the
wire , in order that it stretches the same for 3 times the pulling
weigtht.