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Stiffness of a close-coiled helical spring of wire diameter d, modulus of rigidity G number of coils n, mean coil radius R, is
- a)Gd4/64nR3
- b)Gd4/16nR3
- c)Gd4/8nR3
- d)None of these
Correct answer is option 'A'. Can you explain this answer?
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Stiffness of a close-coiled helical spring of wire diameter d, modulus...
Stiffness of a Close-Coiled Helical Spring
Stiffness or spring constant of a close-coiled helical spring is defined as the force required to produce unit deflection in the spring. It is denoted by k and is given by the formula:
k = (Gd⁴/8nD³) or (Gd⁴/64nR³)
where,
G = modulus of rigidity of the material of the spring
d = diameter of the wire
n = number of coils
D = mean diameter of the spring
R = mean radius of the spring
Explanation
The stiffness of a close-coiled helical spring is directly proportional to the modulus of rigidity of the material of the spring, the fourth power of the wire diameter, and the cube of the mean diameter of the spring. It is inversely proportional to the number of coils and the cube of the mean radius of the spring.
The first formula, k = (Gd⁴/8nD³), is used when the mean diameter of the spring is known. The second formula, k = (Gd⁴/64nR³), is used when the mean radius of the spring is known.
The correct answer is option 'A', i.e., Gd⁴/64nR³. This formula is derived from the formula for the potential energy stored in a spring, which is given by:
U = 1/2 kx²
where,
U = potential energy stored in the spring
k = stiffness or spring constant of the spring
x = deflection of the spring
By differentiating this formula with respect to x, we get the formula for the force required to produce unit deflection in the spring, which is given by:
F = kx
Substituting the value of k from the formula mentioned above, we get:
F = (Gd⁴/8nD³) x or F = (Gd⁴/64nR³) x
Hence, the correct answer is option 'A', i.e., Gd⁴/64nR³.
Stiffness or spring constant of a close-coiled helical spring is defined as the force required to produce unit deflection in the spring. It is denoted by k and is given by the formula:
k = (Gd⁴/8nD³) or (Gd⁴/64nR³)
where,
G = modulus of rigidity of the material of the spring
d = diameter of the wire
n = number of coils
D = mean diameter of the spring
R = mean radius of the spring
Explanation
The stiffness of a close-coiled helical spring is directly proportional to the modulus of rigidity of the material of the spring, the fourth power of the wire diameter, and the cube of the mean diameter of the spring. It is inversely proportional to the number of coils and the cube of the mean radius of the spring.
The first formula, k = (Gd⁴/8nD³), is used when the mean diameter of the spring is known. The second formula, k = (Gd⁴/64nR³), is used when the mean radius of the spring is known.
The correct answer is option 'A', i.e., Gd⁴/64nR³. This formula is derived from the formula for the potential energy stored in a spring, which is given by:
U = 1/2 kx²
where,
U = potential energy stored in the spring
k = stiffness or spring constant of the spring
x = deflection of the spring
By differentiating this formula with respect to x, we get the formula for the force required to produce unit deflection in the spring, which is given by:
F = kx
Substituting the value of k from the formula mentioned above, we get:
F = (Gd⁴/8nD³) x or F = (Gd⁴/64nR³) x
Hence, the correct answer is option 'A', i.e., Gd⁴/64nR³.
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