Final answer:
The inertia of pulleys must be accounted for when calculating a mass-spring system's natural frequency. The effective mass from the inertia of each pulley is half of the pulley's mass, which must be added to the main mass to correct the natural frequency calculation.
Step-by-step explanation:
To understand how the inertia of pulleys affects the natural frequency of a mass-spring system, one must first recognize that the rotation of the pulleys adds to the inertia of the system. This is because the inertia of a rotating body (such as a pulley) contributes to the overall inertia of the system, which affects the natural frequency. To calculate the corrected natural frequency, one must add an effective mass to account for the inertia of the pulleys. Brushing aside friction and damping, the rotational inertia (moment of inertia) of each pulley can be converted to a translational equivalent that can then be added to the mass of the system.
If m1 and m2 represent the masses of uniform disk pulleys and r1 and r2 represent their radii respectively, the moment of inertia for each would be I1 = (1/2)m1r12 and I2 = (1/2)m2r22. The effective mass (meff) for each pulley contributing to the system's mass would be meff1 = I1/r12 = (1/2)m1 and meff2 = I2/r22 = (1/2)m2.
Thus, the total mass used to calculate the natural frequency should be M + meff1 + meff2, where M is the original mass of the main system. The natural frequency is given by the equation f = (1/2π)sqrt(k/(M + meff1 + meff2)), where k is the spring stiffness.