Question

What happens to the frequency of a mass attached to spring when energy becomes 16times?

Answer 1

For an oscillating mass-spring system, the energy balance is:

• `underbrace(E)_("Total Energy") = underbrace(T)\_("Kinetic Energy") + underbrace(U)\_("Potential Energy") = " const"`

Specifically:

• `E = 1/2 mdot x ^2 + 1/2 kx^2`

For a given total energy, `E`, differentiate the lot wrt `t`:

`0 = m cancel(dot x) ddot x + k x cancel(dot x)`

• `implies ddot x + omega^2 x = 0, qquad omega = sqrt(k/m) [= 2 pi f]`

Which solves as the usual SHM:

• `x = A cos (omega t + phi)`

IOW, the (angular) frequency of the system is a property depending only on the mass and spring stiffness.

Increasing the amplitude of the oscillation will increase the energy the system contains, but the system will oscillate with the same frequency.