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Two closed coil springs are made from the same small diameter wire, one wound on 10 cm diameter core and the other on 5 cm diameter core. If each spring had N coils, then the ratio of their spring constants should be
  • a)
    1/16
  • b)
    1/8
  • c)
    1/4
  • d)
    1/2
Correct answer is option 'B'. Can you explain this answer?
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Two closed coil springs are made from the same small diameter wire, on...
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Two closed coil springs are made from the same small diameter wire, on...
Solution:

Given data:

Diameter of larger coil (D1) = 10 cm
Diameter of smaller coil (D2) = 5 cm
Number of coils in each spring (N) = Constant

Formula used:

Spring constant (k) = (Gd⁴/8D³n) × (N² - 1)

where,
G = Shear modulus of material of wire
d = Diameter of wire
D = Mean diameter of spring coil
n = Number of turns or coils
N = Constant

Calculation:

Since both the springs are made from the same small diameter wire, the diameter of wire (d) is constant for both the springs.

Let us consider the spring wound on the 10 cm diameter core as spring 1 and the spring wound on the 5 cm diameter core as spring 2.

Mean diameter of spring coil for spring 1 (D1) = 10 cm
Mean diameter of spring coil for spring 2 (D2) = 5 cm

Since the number of coils in each spring (N) is constant, we can write:

N1 = N2 = N

Now, spring constant (k) can be calculated using the above formula as:

k1 = (Gd⁴/8D1³n) × (N² - 1)
k2 = (Gd⁴/8D2³n) × (N² - 1)

Dividing k1 by k2, we get:

k1/k2 = (D2/D1)³
k1/k2 = (5/10)³
k1/k2 = 1/8

Therefore, the ratio of spring constants of spring 1 and spring 2 is 1/8.

Hence, the correct option is (b) 1/8.
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Community Answer
Two closed coil springs are made from the same small diameter wire, on...
If there are different diameters then the ratio will be same or different
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Two closed coil springs are made from the same small diameter wire, one wound on 10 cm diameter core and the other on 5 cm diameter core. If each spring had N coils, then the ratio of their spring constants should bea)1/16b)1/8c)1/4d)1/2Correct answer is option 'B'. Can you explain this answer?
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