Analysis: That is, find the period T of the simple harmonic vibration.
Answer: Set the length of the compression spring shape variable to x. From Hooke’s law, the elastic force (that is, the restoring force) of the compression spring received by the block is F=kx. (Here only consider the case within the elastic limit of the compression spring)
It can be seen from the above formula that the restoring force of a block is proportional to its displacement. Therefore, the motion of the block is simple harmonic vibration.
There is such a conclusion: the motion of the projection of the mass point on the horizontal plane with a uniform circular motion in the vertical plane is a simple harmonic vibration. (The conclusion drawn by the rotation vector method, you can try to prove it)
Using the above conclusions, it can be seen that the block can be regarded as a projection of an object moving in a uniform circular motion in a vertical plane. They have the same period. The orbit radius of an object moving in a uniform circular motion is x. The speed is equal to The maximum moving speed of the block (think about why), set to v. From the kinetic energy theorem, for the block: (1/2)kx^2=(1/2)mv^2, we get v=x under the root sign (k/m). For objects moving in a uniform circular motion, the orbit length s=2πx, so T=s/v=2π under the radical sign (k/m)
means that the block returns to its original position after 2π root sign (k/m) seconds.
Regarding the formula T=2π (k/m), there are two points to note:
1. From the formula, T has nothing to do with the size of x
2. Unlike the period formula of a simple pendulum T=2π (L/g) is an approximate formula, the formula T=2π (k/m) is an accurate formula. In fact, there are simple pendulums When doing small amplitude vibration, it is similar to the condition of simple harmonic vibration, and the period formula of a simple pendulum can be derived from this formula.