Swing Process Analysis

We take the system composed of people, swings and the earth as the research object, so that in the whole process of swinging the swing, the external force on the system is only the restraint reaction force of the suspension point, the value of which is the same as the cycloid tension T, which is a variable force . However, since the suspension point is fixed, the external force does not work. Gravity is a conservative force, and the force that causes a person to squat and stand up is a non-conservative internal force. According to the functional principle: "The sum of the work done by all external forces and non-conservative internal forces is equal to the increment of the mechanical energy of the particle system." Because the external force does not perform work, there is Aλ=ΔE. Let's study graphs. The first swing of the variable-length pendulum model shown.

a-d: Human body squat: because va=vd=0, so

(1)

d-e: Free swing, the human body is not deformed, and the mechanical energy of the system is conserved.

e-b: The human body is standing, and therefore the moment of gravity and tension on the suspension point O is 0, so the moment of momentum is conserved.

Note that in this process, part of the work done by the human body is converted into the gravitational potential energy of the system, and the other part is converted into energy. The kinetic energy increment is related to the initial swing angle θ0, and the larger the θ0, the greater the kinetic energy increment.

b-c: Free swing, no deformation of the human body, conservation of mechanical energy of the system, and vc=0.

The work done by the non-conservative force of the human body during the whole a-d-e-b-c process (i.e. the first swing):

In the above formula, Δh1 is the height of the center of mass rising after the first swing. According to the functional principle:

Obviously, the second swing c-f-e-b-a is completely similar to the first swing.

It can be seen that the height raised by each swing is related to the length of the cycloid, the amplitude of the change of the center of mass and the initial declination of the swing. When l0 and l1 are constant, it is only determined by the initial declination of the swing. The larger the initial declination angle, the more positive work the human body does in the process of standing up in the equilibrium position, and thus the higher the ascending height. When θn-1>π/2, the internal force also does positive work when the body is contracted, so Δh is larger. If the cycloid is changed to a rigid light rod, with the increase of the number of swings, θ≥π, and then the system It will make a circular motion without swinging back and forth. When θ0=0, Δh1=0. That is, if the initial position is in the vertical position, the swing cannot swing.

The above discussion does not consider the influence of factors such as air resistance during the swing, but these factors are unavoidable in the actual process. Therefore, as long as Aλ>A resistance during each swing, the swing will be higher and higher; when Aλ=A resistance, the system will oscillate with equal amplitude. Obviously, the technical essentials of swinging the swing should be: the human body is fast at the maximum declination angle. Squat down, stand up quickly at the lowest position, and make the initial angle θ0 of the first swing as large as possible.