The spring constant k for a mass-spring system can be expressed in terms of its period T and mass m using the formula:
k = rac{4 heta^2 m}{T^2}
Where:
- k is the spring constant.
- T is the period of the oscillation.
- m is the mass attached to the spring.
To delve into it, the period T of a simple harmonic oscillator, like a mass-spring system, is given by:
T = 2 heta imes rac{ ext{sqrt}(m)}{k}
This equation indicates that the period T depends on both the mass and the spring constant. Rearranging this formula to solve for k gives us the formula stated above. Essentially, a heavier mass or a softer spring (lower k) will result in a longer period of oscillation, while a lighter mass or a stiffer spring (higher k) will result in a shorter period.