A simple pendulum is a device that can produce reciprocating oscillations. A single pendulum is formed by suspending one end of a weightless thin rod or an inextensible thin flexible rope to a certain point in the gravity field, and consolidating a heavy ball at the other end.If the ball is limited to swing in the vertical plane, it is a plane simple pendulum; if the ball is not limited to swing in the vertical plane, it is a spherical simple pendulum.
The approximate period formula of simple pendulum motion is:T=2π√(L/g)。Among them,LIs the pendulum length,gIs the local gravitational acceleration.
At very small amplitudes (angles),Simple pendulum in simple harmonic motionPeriod following pendulum lengthsquare rootIs proportional to the square root of gravity acceleration, is inversely proportional to the amplitudeSwing the ballThe quality of is irrelevant.
formula
The simple pendulum is an ideal physical model, which is composed of an idealized pendulum ball and a cycloid;The density of the pendulum ball is large, and the radius of the ball is much smaller than the length of the cycloid, so that the pendulum ball can be regarded as a particle, and a simple pendulum is composed of a cycloid and a pendulum ball. Under the condition that the deflection angle is less than 10 °, the period of the simple pendulum is
Schematic diagram of force analysis of simple pendulum in one vibration cycle
It can be seen from the formula that the period of a simple pendulum is independent of the amplitude and the mass of the pendulum.From the perspective of force, the restoring force of a simple pendulum is the component of gravity along the tangent direction of the arc and pointing to the equilibrium position. The larger the deflection angle, the greater the restoring force, the greater the acceleration (gsin θ), and the greater the arc length traveled in the same time. Therefore, the period is independent of amplitude and mass, but only related to the pendulum length l and gravity acceleration g.In some vibration systems, l may not be the rope length, and g may not be 9.8m/
Therefore, the problem of equivalent pendulum length and equivalent gravity acceleration arises.
In physics, some problems are similar to the simple pendulum. After some equivalence, the periodic formula of the simple pendulum can be applied. This kind of problem is called "equivalent simple pendulum". The equivalent simple pendulum is relatively common in life. In addition to the equivalent simple pendulum, the simple pendulum model is also used in other problems
explain
Change of velocity and acceleration of simple pendulum in a vibration period
Simple pendulum is a kind of particle vibration system, which is the simplest pendulum.An object swinging back and forth around a suspension point is called a pendulum, but its period is generally related to the shape, size and density distribution of the object.However, if a small mass is suspended at one end, the fixed length islOn the string that cannot be extended, pull the mass away from the balance position so that the angle between the string and the plumb line passing through the suspension point is less than 10 °. After letting go, the mass vibrates back and forth, which can be regarded as the vibration of the particleTAnd length onlylAnd local gravity accelerationgRelated, i.eTIt has nothing to do with the mass, shape and amplitude of the mass. Its motion state can be expressed by simple harmonic vibration formula, which is called simple pendulum.If the angle of vibration is greater than 10 °, the period of vibration will increase with the increase of amplitude, and it will not be a simple pendulum.If the size of the pendulum ball is quite large and the mass of the rope cannot be ignored, it becomes a compound pendulum, and the period is related to the size of the pendulum ball.
application
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one
Gravity acceleration measurementThe local gravity acceleration can be calculated by measuring the pendulum length and period, which is of great significance to geophysics, geological exploration and other fields.
two
Time and frequency measurement: With the isochronity of simple pendulum, various timers and clocks can be made, such as mechanical pendulum clock, electronic clock, etc., for measuring time and frequency.When the period of simple pendulumT=At 2s, according to the formula, the pendulum length is about 1m, and the simple pendulum in this case is called the second pendulum.
Schematic diagram of second pendulum
Note: In the current high school stage, it is generally studied that the swing angle is less than 10 ° (that is, it is approximately regarded as simple harmonic motion), and the high school teaching materials only involve the extrapolation formula in the experiment, not the derivation of the simple pendulum period formula (because it needs to involve advanced mathematics).Measuring the acceleration of gravity with a simple pendulum is an important application of the simple pendulum period formula.[1]
kinetic equation
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According to Newtonian mechanics, the motion of a simple pendulum can be described as follows.
First, we can get that the torque of gravity on the pendulum is
Where m is the mass, g is the acceleration of gravity, l is the length of the pendulum, and θ is the included angle between the simple pendulum and the vertical direction. Note that θ is a vector, and its projection in the positive direction is taken here.
We hope to get the function of pendulum angle θ with respect to time to describe the motion of a simple pendulum.According to the angular momentum theorem,
among
Is the moment of inertia of the simple pendulum,
Is the angular acceleration.
So the simplification results in
（1）
Small angle approximate period
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(1) Equation is a nonlinear differential equation.So strictly speaking, the motion of the simple pendulum described in equation (1) above is not simple harmonic motion.
However, when θ is small, there is approximately sin θ≈ θ.(i.e
。）Thus, equation (1) becomes
, which is a second-order linear homogeneous differential equation with constant coefficients, and its general solution is
, where A
Is an arbitrary constant, given by the initial value condition.and
So the nonlinear motion of the simple pendulum is linearly approximated to simple harmonic motion
In the college entrance examination and other examinations, it is generally believed that the degree below 10 ° can be approximated in this way.
In fact, 5 ° ≈ 0.087266 rad and sin 5 ° ≈ 0.087155, the difference between the two is only one thousandth of a point, which is very close.In low precision experiments, this systematic error can be ignored (because the accidental error in experimental operation is larger than it).However, if it is changed to 25 °, the error is as high as 3%, so it is no longer considered as simple harmonic vibration.
Due to the nature of the sine function, this approximation is more accurate when the angle is smaller, and less accurate when the angle is larger.If the angle is very large (for example, at 60 degrees, the error is as high as 17%), it can not be said to be a simple harmonic vibration at all.
Comparison between small angle approximate formula and actual curve
Galileo was the first to discover the isochronity of the vibration of a pendulum, and used experiments to find out that the period of a simple pendulum changes with the quadratic root of its length.Huygens made the first pendulum clock.The simple pendulum is not only an instrument for accurately measuring time, but also can be used to measure the change of gravitational acceleration.J. Richel, a contemporary astronomer of Huygens, once brought the pendulum clock from Paris to French Guiana in South America, and found that it was 2.5 min slower every day, and after calibration, it was 2.5 min faster when it returned to Paris.Huygens concluded that this was due to the weakening of gravity caused by the rotation of the earth.1. Newton used a simple pendulum to prove that the weight of an object is always proportional to its mass.Until the middle of the 20th century, the pendulum was still the main instrument for gravity measurement.
True cycle derivation
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The above mentioned is the approximate formula of the simple pendulum when the angle is relatively small, but the science is rigorous, so the periodic formula of the simple pendulum at any angle is supplemented here.
Before that, two concepts (Mathematica's definition is used here) are proposed:
Incomplete elliptic integral of the first kind:
Complete elliptic integral of the first kind:
The following is a discussion with differential equations. We can try to use the kinetic energy theorem to calculate, and we can get its special solution more concisely.
Let the length of the pendulum be l, and the included angle between the cycloid and the vertical direction be θ, then the motion formula of the simple pendulum is:
order
, so there is
The above formula is rewritten as:
This is a differential equation with separable variables!Detach variables:
The general solution is
Given initial conditions
（0≤α≤π），
, the special solution is:
So consider t (t is a quarter cycle):
set up
, then
Considering that
Can be simplified to get
According to the previous definition
Here, α is the pendulum angle.
Related differences
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The visual difference diagram is shown on the right:
Periodic difference
Use computer software to list the difference between approximate formula and real formula.
The following data are all relative errors: relative error=(true value - approximate value)/true value
For each line, the swing angle differs by 1 degree from 0 to 180 degrees.
