Question

The spring described by u'' + 9u = sin(3t) experiences resonance. If the velocity goes above 15 cm per second or below -15 meters per second, the spring breaks. When will the spring break if u(0) = 0 and u'(0)?

Answer 1

The problem involves determining the point in time at which a spring in SHM breaks due to excessive velocity, caused by resonance. The lack of specific information for the initial velocity makes it impossible to calculate the exact time of breaking.

Step-by-step explanation:

The question involves a spring system undergoing simple harmonic motion (SHM) and is based on the concept of resonance in a mechanical oscillator.

The differential equation given, u'' + 9u = sin(3t), typically describes the motion of a mass attached to a spring in the presence of an external periodic force. According to the question, if the velocity of the mass exceeds 15 cm/s or goes below -15 m/s, the spring breaks.

Given the initial conditions u(0) = 0 and an unspecified initial velocity u'(0), we would generally have to solve the differential equation to determine the function u(t), which describes the position of the mass over time, and then find its derivative to determine the velocity u'(t). When u'(t) reaches 15 cm/s or -15 m/s, the spring will break.

However, since the exact initial velocity is not specified, we cannot calculate the exact moment at which the spring would break.