In the process of swinging, according to the law of rotation:
Simplification and arrangement
Make, multiply both sides of the equation by dq, and then rewrite it into differential form, and there is an integral formula, where c is an integral constant, which can be written from the initial conditions. Let t=0, and then substitute it into the above formula to find (1)(2).
When t changes from t=0, q changes from q = Q0 to q = 0, and (1) and (2) should be signed. For the integral of (2), a new variable Ф is introduced in (3) to make (4)(5).
Then (6)
Derive both sides of formula (4) to get (7)
Substituting the formulas (6) and (7) into the formula (3) is obtained.
Expand denominator with binomial and integrate item by item.
therefore
Substitute the values of w and k to get (8).
The above formula is the periodic formula of a simple pendulum. The formula shows that the larger the swing amplitude q0, the longer the period. If it is made, it can be calculated as t =1.04t0; T= 1. 17T0。 When q0 is very small (in general), you can omit the high-order term and get. This is the periodic formula of small amplitude swing of a simple pendulum.
A particle vibration system 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 on a string with a fixed length of L at one end, which cannot be extended, and the mass is pulled away from the equilibrium position so that the included angle between the string and the vertical line of the suspension point is less than 10, the reciprocating vibration of the mass can be regarded as the vibration of a particle, and its period t is only related to L and the local gravity acceleration G, that is, it has nothing to do with the mass, shape and amplitude of the mass, and its motion state can be simple and harmonic. If the angle of vibration is greater than 10, the period of vibration will increase with the increase of amplitude, not a simple pendulum. If the size of the pendulum ball is quite large and the quality of the rope cannot be ignored, it becomes a compound pendulum (physical pendulum), and the period is related to the size of the pendulum ball.