Springs must be familiar to everyone
Compared with wire or other materials of rope, what are the characteristics of springs?
Compared to wire, springs have better toughness and elastic deformation limits in terms of bending
And the spring is not as loose as a rope, and is able to transmit the force well to each of its positions
Therefore, the spring is a good helper for teachers in many teaching experiments!
Today we're going to play something more interesting
Experimental equipment
Cups, springs, gloves, emblems
Experimental process
First, let's do a review:
Hang the emblem on one end of the spring
(Compared to springs, the mass of the emblem is negligible)
Let it swing inside the plane
At this point, the spring can be approximately regarded as a compound pendulum
The cycle and equivalent length of the swing motion are positively correlated
As the length of the spring increases, as the moment of inertia increases,
The reciprocating cycle has increased
If we superimpose reciprocating motions within two vertical planes
It is not possible to synthesize a circular motion
This circular motion has a lower angular velocity limit
This lower limit is related to the frequency of oscillation before the formation of a circular motion
But on top of that we can try to make it turn faster
Relative springs also "float" higher
After reviewing, let's start playing:
Pinch one end of the spring
Make it spin
Not only can you get the simple circular motion in the above figure
You can also get motion similar to standing waves
Achieve this minimum angular velocity condition for the standing wave
It is related to the natural frequency of the spring oscillating at a specific length
Can you guess what is the ratio of this length to the total length of the spring?
(Let me see where the witty little friend is)
Again, after the formation of a "standing wave"
If we accelerate the rotation
The spring as a whole also lifts
The feel of turning is still very good
Compared with the standing wave in the plane, the rotating "standing wave" is more stable
You can also try vibrating in only one direction
As a result, it will be difficult to control the swing within a plane
The spring will naturally start to turn
So, can we make the spring appear more joints?
We can first try to find the frequency that stabilizes the vibration
First find 1/5 of the spring (think why?). ）
Shake it gently
Calm down and feel its vibrations
The spring is then shaken at the same frequency
You can successfully inspire more knots
Look, a spring that rotates like this doesn't look like a snake dancing to the music?
Explanation of the principles
(1) About the pendulum
The period formula for swinging at a small angle is:
where I is the moment of inertia of the spring around the pivot point (hand).
If the deformation of the spring is not considered at a small angle
Therefore, at a small angle, the square of the vibration period is proportional to the length of the spring
As the swing angle and spring length increase, the cycle formula and the moment of inertia change, but the overall correlation is positive.
(2) About the compound pendulum movement and the circular motion
At small angles, the complex pendulum approximates the simple harmonic motion, and in the two directions of orthogonality, two simple harmonic motions with a phase difference of π/2 can be combined into a circular motion. You can refer to this pair of parameter equations:
Of course, as the angular velocity of the circular motion increases, the entire system cannot be synthesized with a simple harmonic motion approximated by a complex pendulum. At this time, the circular motion is balanced by the centrifugal force of the spring, gravity and the tensile force inside the spring. Of course, since the spring has a curve when it is shaken, some calculus calculations are required to get the complete expression. Of course, we can qualitatively analyze that as the angular velocity of the circular motion increases, the centrifugal force on the spring must increase, and its combined force with gravity and the angle between the vertical axis will also become larger, and the spring as a whole will "float".
(3) About standing waves
We mentioned earlier that two simple harmonic motions can be combined into a circular motion, and the reverse is the same, we can project a rotating spring onto a plane and have it break down into two plane waves. The spring is fixed at one end of the hand and not fixed at the other end, so forming a standing wave is the pattern we learned in middle school when we open and close one end of the tube. Then the first two standing wave diagrams formed are as follows:
This is why we found that the position closest to the end of the spring is 1/3 and 1/5 respectively
Edit: Lychee