TS EAMCET 04 May 2019 Shift 2 Solved Paper

Consider the figure shown below
Let dl be the change in the length of small element due to rotation by the application of force dF
The expression for Young's modulus is given by,
Y‌‌=‌

dF∕A

dl∕x

dF‌‌=‌

AYdl

x

‌‌‌‌‌⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅(I)
Now, dF is the centrifugal force due to mass of the element of length dx
dF‌‌=dm⋅xω2
‌‌=‌

m

L

⋅xdxω2‌‌‌‌‌⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅(II)
From equation (I) and (II),
‌

AYdl

x

=‌

m

L

xdxω2
dl=‌

mx2ω2

LAY

dx
Integrate the above equation
∫dl‌‌=

L

∫

0

‌

mx2ω2

LAY

dx
l‌‌=‌

mL2ω2

3AY

=(‌

m

AL

)(‌

ω2L3

3Y

)
=(‌

ρω2L3

3Y

)