The ratio of deflection in spring A to B is 1/8. The correct answer is option 1. 1/8.
Here's why:
Spring Deflection Formula: The deflection of a helical spring is directly proportional to the applied load (W) and inversely proportional to the spring constant (k). It can be expressed by the formula:
δ = W / k
Spring Constant and Mean Diameter: The spring constant of a helical spring is inversely proportional to the cube of the mean diameter (D) and can be expressed by the formula:
k = Gd⁴ / 8D³n
where:
G is the modulus of rigidity of the spring material
d is the diameter of the wire
n is the number of active coils
Given Information: We are given that both springs have the same number of active coils (n) and the same wire diameter (d). The mean diameter of spring A (DA) is half that of spring B (DB). This means DA = DB/2.
Comparing Spring Constants: Using the formula for spring constant, we can compare the spring constants of A and B:
KA = GBd⁴ / 8(DA)³n
KB = GBd⁴ / 8(DB)³n
Dividing these two equations, we get:
KA / KB = (DB)³ / (DA)³ = (2DA)³ / DA³ = 2³ = 8
Therefore, the spring constant of spring A is 8 times higher than the spring constant of spring B.
Ratio of Deflections: Using the formula for deflection and substituting the information we have:
δA / δB = (KB / KA) = 1 / 8
Option 1 is the right choice.
Question:-
If two closely coiled helical springs A&B with the mean diameter of spring A is half of that of spring B and having equal number of active coils and same wire diameter are subjected to same axial load of W., then the ratio of deflection in spring A to B.
1. 1/ 8
2. 1/ 4
3. 2
4. 8