A detailed introduction to the principle of isochronism of the pendulum

The principle of isochronism of the pendulum means that the time to complete a swing is the same regardless of whether the swing amplitude (when the swing angle is less than 5°) is large or small. ”

It is generally believed that Galileo discovered this principle, and he came to this conclusion while observing the phenomenon of the swinging chandelier in the church of Pisa. According to the principle of isochronism, if the amplitude of the pendulum is small, the period of the pendulum is independent of the amplitude of the pendulum. Although isochronism was known to the Arabs centuries before Galileo, Galileo was the first scientist to study the phenomenon with a rigorous scientific attitude.

He pointed out that the period of the pendulum does not depend on the number of objects hanging on the cycloid, but only on the square root of the length of the cycloid. If the effect of resistance is not taken into account, the law of oscillation of a cork ball or shot put ball suspended on an equidistant line is the same.

The principle of isochronism of the pendulum means that the time to complete an oscillation is the same regardless of whether the swing amplitude (when the swing angle is less than 5°) is larger or smaller.

It is widely credited that Galileo Galilei discovered this principle, which he drew when he observed the phenomenon of chandelier swinging in a church in Pisa. According to the principle of isochronism, if the amplitude of the pendulum is small, then the period of the oscillation is independent of the amplitude of the oscillation. Although isochronism was well known to the Arabs for centuries before Galileo, Galileo was the first to be recommended by scientists who studied the phenomenon with a rigorous scientific attitude.

He pointed out that the period of the pendulum does not depend on the amount of hanging on the cycloid, but only on the square root of the cycloid length. If the effect of drag is not taken into account, the law of oscillation of a cork ball or a lead ball suspended on an isometric line is the same.

Frequency increase: Pull one end of the cycloidal activity to shorten the pendulum length, and the frequency of the pendulum will increase.

Gently push the pendulum to make it oscillate at a small amplitude, then pull the moving end of the cycloid, shorten the length of the pendulum, and you will notice that the frequency of the swing will be faster and faster. If the length of the pendulum is reduced to 1 4, the period of the pendulum is reduced by 1 2 times. Of course, if you want to get accurate data, you'll need to take dozens of measurements of the swing time.

It is isochronism using the single pendulum. It is this property that can be used for timing.

The periodic formula of the single pendulum is: time = 2 times the pi multiplied by (the length of the root pendulum divided by the acceleration of gravity) Through the formula and its derivation, it can be seen that the movement of the single pendulum depends on gravity and the pull of the rope. The period of the swing depends only on the length of the rope and the acceleration due to gravity.

The Earth's gravitational acceleration is fixed, and controlling the pendulum length allows the period to be adjusted to time it.

If a particle starts from any point of this section of the cycloid and slides down along the cycloid under the action of gravity, the time required for this particle to reach the lowest point c has nothing to do with the position of the starting point. That is, they arrive at point c at the same time from any two points of difference.

is (a g), where a is the radius of the moving circle corresponding to the cycloid.

After Huygens invented the pendulum clock in 1657 using the principle of isochronism of a single pendulum, he gradually discovered that the isochronism of a single pendulum is limited—that is, when the pendulum angle is less than 5°, the sinusoidal value of the pendulum angle can be approximated as the value of the pendulum angle, that is, sin and thus has a period t=2 (l g).

After continuous research, it was found that the isochronism of the cycloid was not affected by the angle of the pendulum, so in 1673 the pendulum clock with true isochronism was made by using the isochronism of the cycloid.

Regardless of the amplitude, the period is a fixed value, which is equal to 4a g).

It is proved that the parametric equation of the reversed cycloid is x=a -asin , y=-a+a*cos, and the parameter corresponding to the starting point p of the particle slide is 0< When the particle slides to the point of parameter , according to the law of conservation of energy, the potential energy lost by the particle is converted into kinetic energy, so the instantaneous velocity of the particle at that point is v( )= (2ag(cos -coa)).

On the other hand, the differentiation of the arc length s is ds= ((dx) +dy) )=2a*sin( 2)d

Thus, the time it takes for the particle to slip to the lowest point c is [2a*sin( 2)d ) (2ag(cos-coa))).

This value is equal to a g), regardless of .

For a pendulum of the same length (i.e., the same length of rope), no matter what the distance from the lowest point (no more than a quarter of the circle) the time to reach the lowest point is the same.

Pendulum. Isochronism.

A period is the time that a pendulum experiences twice in a row through a certain point in the same way, for example, the time it takes for a pendulum to reach the highest point on one side and the next time it returns to that point.

Swing cycle. And not from the swing to the end of it completely.

The time of the stop.

Of course, the time it takes from the time it starts to swing to the time it stops, divided by the number of swings in the process, is equal to the time it takes to swing each time, which is also quite its "cycle".