Solution to quality problems of compression spring

Analysis: that is to find the period T of simple harmonic vibration
Answer: let the shape variable length of the spring be X. according to Hooke’s law, the elastic force (i.e. restoring force) of the compression spring on the object block is f = KX. (only the case within the elastic limit of the compression spring is considered here.)
According to the above formula, the restoring force of the block is proportional to its displacement
There is a conclusion: the motion of the projection of a particle in a vertical plane with uniform circular motion on the horizontal plane is a simple harmonic vibration
Using the above conclusion, we can see that the block can be regarded as the projection of an object moving in a uniform circular motion in the vertical plane. They have the same period. The radius of the orbit of the object moving in a uniform circular motion is X. the size of the velocity is equal to the maximum velocity of the block (think about why). Let v, For the object block: (1 / 2) KX ^ 2 = (1 / 2) MV ^ 2, we get v = x root (K / M). For a body moving in a uniform circle, its orbit length is s = 2 π x, so t = s / v = 2 π x (K / M)
That is to say, the block returns to its original position after 2 π root sign (K / M) seconds
For the formula t = 2 π root sign (K / M), there are two points to note:
1. According to the formula, t has nothing to do with the size of X
2. Unlike the periodic formula of a simple pendulum, t = 2 π radical (L / g) is an approximate formula, but the formula t = 2 π radical (K / M) is an accurate formula. In fact, there are conditions under which a simple pendulum is approximate to a simple harmonic vibration when it vibrates in small amplitude