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Two helical tensile springs of the same material and also having identical mean coil diameter and weight, have wire diameters d and d/2. The ratio of their stiffness is: 
  • a)
    1
  • b)
    4
  • c)
    64
  • d)
    128
Correct answer is option 'C'. Can you explain this answer?
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Given:
Two helical tensile springs of same material
Identical mean coil diameter and weight
Wire diameters d and d/2

To find:
Ratio of their stiffness

Solution:
Stiffness of a spring is given by the formula:
k = Gd⁴/8D³n
where,
k = stiffness
G = modulus of rigidity
d = wire diameter
D = mean coil diameter
n = number of active coils

Since both the springs have identical mean coil diameter and weight, we can assume that they have same number of active coils.

Let the stiffness of the spring with wire diameter d be k1
Let the stiffness of the spring with wire diameter d/2 be k2

Then, using the formula for stiffness, we have:
k1 = Gd⁴/8D³n
k2 = G(d/2)⁴/8D³n

Dividing k1 by k2, we get:
k1/k2 = (Gd⁴/8D³n)/(G(d/2)⁴/8D³n)
k1/k2 = (d⁴/16)/(d⁴/2⁴)
k1/k2 = 16/1
k1/k2 = 64

Therefore, the ratio of stiffness of the two springs is 64.

Answer: Option C (64)
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Two helical tensile springs of the same material and also having identical mean coil diameter and weight, have wire diameters d and d/2. The ratio of their stiffness is:a)1b)4c)64d)128Correct answer is option 'C'. Can you explain this answer?
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