A rod of length 1.05 m having negliaible mass is supported at its ends by two wires of steel (wire A) and aluminium (wire B) of equal lengths as shown in fig. The cross-sectional area of wire A and B are `1 mm^(2)` and 2` mm^(2)`, respectively . At what point along the rod should a mass m be suspended in order to produce (a) equal stresses and (b) equal strains in both steel and aluminium wires. Given,
`Y_(steel) = 2 xx 10^(11) Nm^(-2) and Y-(alumi n i um) = 7.0 xx 10^(10)N^(-2)`

Options

A. Mass m should by suspended close to wire A to have equal stresses in both the wires
B. Mass m should be suspended close to B to have equal stresses in both the wires
C. Mass m should be suspended at the middle of the wires to have equal stresses in both the wires
D. Mass m should be suspended close to wire A to have strain in both wires

Answer

B, D

Explanation

Let the mass is placed at x from the end B.

Let `T_(A)` and `T_(B)` be the tensions in wire A and wire B respectively.
For the rotational equilibrium of the system,
`" "Sigmatau=0" (Total torque = 0)"`
`implies " "T_(B)x - T_(A)(l-x)=0`
`implies " "(T_(B))/(T_(A)) = (l-x)/(x) " ....(i)"`
Stress in wire `A = S_(A) = (T_(A))/(a_(A))`
Stress in wire `B=S_(B) = (T_(B))/(a_(B))`
where `a_(A)` and `a_(B)` are cross-sectional areas of wire A and B respectively.
By equation `a_(B) = 2 a_(A)`
Now, for equal stress `S_(A) = S_(B)`
`implies" " (T_(A))/(a_(A))=(T_(B))/(a_(B))implies (T_(B))/(T_(A))=(a_(B))/(a_(A))=2`
`implies " "(l-x)/(x)=2 implies (l)/(x)-1=2`
`implies " "x=(l)/(3) implies l-x = l-l//3= (2l)/(3)`
Hence, mass m should be placed closer to B.
For equal strain, `" "("Strain")_(A)=("Strain")_(B)`
`implies ((Y_(A)))/(S_(A))=(Y_(B))/(S_(B))` (where `Y_(A)` and `Y_(B)` are Young modulii)
`implies (Y_("steel"))/(T_(A)//a_(A))=(Y_(Al))/(T_(B)//a_(B))`
`implies " "(Y_("steel"))/(Y_(Al))=(T_(A))/(T_(B))xx(a_(B))/(a_(A))=((x)/(l-x))((2a_(A))/(a_(A)))`
`implies " "(200xx10^(9))/(70xx10^(9))=(2x)/(l-x)implies (20)/(7)=(2x)/(l-x)`
`implies " "(10)/(7)=(x)/(l-x)implies 10l - 10x = 7x`
`implies " "17x = 10l implies x = (10l)/(17)`
`" "l-x=l-(10l)/(17)=(7l)/(17)`
Hence, mass m should be placed closer to wire A.