A spring of force constant K is cut into two parts in langth ratio 1:3 then the force constant of the small part will be

When a spring is cut into two parts, the force constants of the resulting pieces change based on their lengths. In your case, we have a spring that is divided in a length ratio of 1:3. To find the force constant of the smaller part, we can use the relationship between the length of a spring and its spring constant.

Understanding Spring Constants

The spring constant, denoted as K, is a measure of a spring's stiffness. According to Hooke's Law, the force exerted by a spring is directly proportional to its extension or compression, expressed as F = kx, where F is the force, k is the spring constant, and x is the displacement from its equilibrium position.

Relationship Between Length and Spring Constant

When a spring is cut into two parts, the spring constant of each part is inversely proportional to its length. This means that if you have a spring with a total length L and spring constant K, when you cut it into two parts with lengths L1 and L2, the spring constants K1 and K2 of the two parts can be calculated using the following relationships:

• K1 = K * (L / L1)
• K2 = K * (L / L2)

Calculating the Spring Constants

In your scenario, the spring is divided in a ratio of 1:3. Let's denote the total length of the spring as L. The lengths of the two parts will be:

• Length of the smaller part (L1) = L/4
• Length of the larger part (L2) = 3L/4

Now, we can find the spring constants for each part. The spring constant for the smaller part (K1) can be calculated as follows:

K1 = K * (L / L1) = K * (L / (L/4)) = K * 4 = 4K

Conclusion

The force constant of the smaller part of the spring, which is one-fourth the length of the original spring, will be four times the original spring constant. Therefore, if the original spring constant is K, the spring constant of the smaller part will be:

This illustrates how cutting a spring into shorter lengths increases its stiffness, resulting in a higher spring constant for the smaller segment. If you have any more questions about springs or related concepts, feel free to ask!