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प्रश्न
Answer the following.
A wire gets stretched by 4 mm due to a certain load. If the same load is applied to a wire of same material with half the length and double the diameter of the first wire, what will be the change in its length?
उत्तर
Given: l1 = 4 mm = 4 × 10-3 m
L2 = `"L"_1/2,` D2 = 2D, r2 = 2r1
To find: Change in length (l2)
Formula: Y = `"FL"/("A"l) = "FL"/(pi"r"^2l)`
Calculation:
From formula,
`"Y"_1 = ("F"_1"L"_1)/(pi"r"_1^2l_1)` .....(i)
`"Y"_2 = ("F"_2"L"_2)/(pi"r"_2^2l_2)` .....(ii)
Dividing equation (ii) by equation (i),
`"Y"_2/"Y"_1 = (("F"_2"L"_2)/(pi"r"_2^2l_2))/(("F"_1"L"_1)/(pi"r"_1^2l_1))` .....(iii)
Since same load is applied on same wire, Y1 = Y2 and F1 = F2
∴ `"L"_1/("r"_1^2 l_1) = "L"_2/("r"_2^2l_2)` .....[From (iii)]
`l_2 = ("L"_2 xx "r"_1^2 xx l_1)/("r"_2^2 xx "L"_1)`
`= ("L"_2 xx "r"_1^2 xx l_1)/(4"r"_1^2 xx 2 xx "L"_2) .....(because "L"_1 = 2"L"_2, "r"_2 = 2"r"_1)`
`= l_1/8`
`= (4 xx 10^-3)/8`
= 0.5 × 10-3 m
= 0.5 mm
The new change in length of the wire is 0.5 mm.
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