The maximum kinetic energy of a mass-spring system in SHM is equal to

(1/2) m Vmax^2

if x = a sin w t
v = a w cos wt
so
Vmax = a w

a is amplitude
w is circular frequency = 2 pi f

so
Ke max = (1/2) m (a^2 w^2)
= 2 pi^2 m a^2 f^2

To find the maximum kinetic energy of a mass-spring system in Simple Harmonic Motion (SHM), we need to consider the relationship between the kinetic energy and the potential energy of the system.

In SHM, the total mechanical energy remains constant. It is the sum of the potential energy (PE) and kinetic energy (KE) of the system. Mathematically, this can be represented as:

Total Mechanical Energy (E) = Potential Energy (PE) + Kinetic Energy (KE)

At the extreme points of the oscillation (i.e., when the displacement is maximum), all the potential energy is converted into kinetic energy. Therefore, at these points, the kinetic energy is at its maximum.

Since the total mechanical energy remains constant throughout the motion, the maximum kinetic energy is equal to the total mechanical energy. In other words:

Maximum Kinetic Energy (KE_max) = Total Mechanical Energy (E)

To determine the maximum kinetic energy, we need to know the formula for the total mechanical energy of a mass-spring system in SHM. This can be calculated using the formula:

Total Mechanical Energy (E) = 0.5 * k * A^2

where:
- k is the spring constant of the system
- A is the amplitude of the oscillation

By plugging in the appropriate values for k and A, you can calculate the maximum kinetic energy (KE_max) of the system.