How is the period of a mass-spring system calculated?

The period of a mass-spring system is calculated using the formula T = 2π√(m/k), where m is the mass and k is the spring constant.

To understand how this formula is derived, we need to first understand the motion of a mass-spring system. When a mass is attached to a spring and displaced from its equilibrium position, it experiences a restoring force that is proportional to the displacement. This is described by Hooke's Law, which states that F = -kx, where F is the restoring force, k is the spring constant, and x is the displacement from equilibrium.

Using Newton's Second Law, we can write the equation of motion for the mass-spring system as m(d^2x/dt^2) + kx = 0, where m is the mass and t is time. This is a second-order differential equation, which can be solved using the characteristic equation r^2 + (k/m) = 0. The solution to this equation is r = ±i√(k/m), which gives us the general solution x = A cos(√(k/m)t) + B sin(√(k/m)t), where A and B are constants determined by the initial conditions.

To find the period of this motion, we need to find the time it takes for the mass to complete one full cycle. This occurs when the cosine function reaches its maximum value of 1, which happens when √(k/m)t = 2π. Solving for t, we get t = 2π√(m/k), which is the formula for the period of a mass-spring system.