Elasticity

Lengths: ‌

lA

lB

=‌

1

2

Diameters: ‌

dA

dB

=‌

3

2

implies ‌

rA

rB

=‌

3

2

. Thus, ‌

rA2

rB2

=‌

9

4

.
Forces: ‌

FA

FB

=‌

3

1

Elastic Potential Energy Per Unit Volume:
The elastic potential energy per unit volume in the wire is given by:
U=‌

1

2

‌

F⋅∆L

V

Young's Modulus Definition:
From Young's modulus Y, we have:
Y=‌

F⋅L

A⋅∆L

⟹∆L=‌

F⋅L

A⋅Y

Volume:
V=A⋅L
Substituting ∆L and V into the expression for U :
U=‌

1

2

⋅‌

F⋅F⋅L

A⋅L⋅A⋅Y

=‌

1

2

‌

F2

A2Y

Ratio of Elastic Potential Energies UA and UB :
‌

UA

UB

=‌

‌ 1 2 FA2 (πrA2)2Y

1 2 ‌ FB2 (πrB2)2Y

=‌

FA2(rB2)2

FB2(rA2)2

Now plug in the ratios:
‌

UA

UB

=(‌

3

1

)2⋅(‌

4

9

)2
Calculate the final ratio:
‌‌

UA

UB

=‌

9

1

⋅‌

16

81

‌‌

UA

UB

=‌

144

81

‌‌

UA

UB

=‌

16

9

Thus, the ratio of elastic potential energies stored per unit volume in wires A and B is 16:9.