A compression spring is made of music wire of 2mm diameter having a shear strength and shear modulus of 800 MPa and 80 GPa respectively. The mean coil diameter is 20 mm, free length is 40 mm and the number of active coils is 10. If the mean coil diameter is reduced to 10 mm, the stiffness of the spring is approximately

decreased by 8 times

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decreased by 2 times

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increased by 2 times

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increased by 8 times

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Solution

The correct option is D increased by 8 times
$k = \frac{G d^{4}}{8 D^{3} N}$

$\frac{k_{2}}{k_{1}} = {(\frac{D_{1}}{D_{2}})}^{3} = {(\frac{20}{10})}^{3} = 8$

$∴$ Stiffness of spring is increased by 8 times.

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