Given g=10 m/sec2 and Young Modulus of the material of the wire=2X1011 N/m2
Stress =Load/Area
=mgA=100×1010−4=107 N/m2
Now
Young Modulus=Stress/Strain
So strain= Stress/Young Modulus
=5×10−5
Now we know that
Strain = Increase in length/Initial length
=> Increase in Length=strain * Initial length
=5×10−5 m
Elastic Potential energy
=12stress×strain×Volume
=5×10−2 J
The tensions in the load position will the weight of the mass
So T=mg
=100*10=1000 N
Let us consider a small element dx at a distance x from the rotating end.
Now mass of the rod =Lπr2ρ
Mass per unit length of the rod=πr2ρ
Mass of the small element=πr2dxρ
Centripetal force acting on the element
dF=πr2ρxω2dx
The Tension in the element would be due to the centripetal force of the outer portion i.e from x=x to x=L
F=∫Lxπr2ρxω2dx
F=12πr2ρω2(L2−x2)
Tension at fixed point
x=0
F=ρπr2ω2L22
At x=L/2
F=3ρπr2ω2L8
So tension is maximum at fixed end and decrease towards outer end.
So
Sπr2>ρπr2ω2L22
Or √2SρL2<ω
Let dy be the elongation at the element of length dx
Then
dydx=1YF(x)A
Substituting the value of F(x) from previous part
dy=πr2ρω2(L2−x2)dx2Yπr2
y=∫L0ρω22Y(L2−x2)dx=ρω2L33Y
Let l2 be the elongation in second case then
T1=2π√l1g ---(1)
T2=2π√l2g ---(2)
So that
T21T22=l1l2
Or
l1l2=1625
Now
For first case
Y=MgLAl1 --(3)
Second case
Y=(M+m)gLAl2 --(4)
From 3 and 4
l1l2=MM+m
Or
MM+m=1625
M+mM=2516
Or
M+m−MM=25−1619
Or
mM=916
Let l1 = AB, l2 = AC, l3 = BC
cosθ=l23+l21−l222l3L1
or
2l3L1cosθ=l23+l21−l22
Differentiating
2(l3dL1+l1dl3)cosθ−2l3L1sinθdθ=2l3dl3+2l1dl1−2l2dl2
Now
dl1=l1α1ΔT
dl2=l2α1ΔT
dl3=l3α2ΔT
and l1=l2=l3=l
Substituting these values
2(l2α1ΔT+l2α2ΔT)cosθ−2l2sinθdθ=2l2α1ΔT+2l2α1ΔT−2l2α2ΔT
sinθdθ=2α1ΔT(1−cosθ)−α2ΔT
Substituting θ=600
dθ=2(α1−α2)ΔT√3