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In physics and mathematics, in the area of dynamical systems, an elastic pendulum [1] [2] (also called spring pendulum [3] [4] or swinging spring) is a physical system where a piece of mass is connected to a spring so that the resulting motion contains elements of both a simple pendulum and a one-dimensional spring-mass system. [2]
A mass m attached to a spring of spring constant k exhibits simple harmonic motion in closed space. The equation for describing the period: = shows the period of oscillation is independent of the amplitude, though in practice the amplitude should be small. The above equation is also valid in the case when an additional constant force is being ...
When a spring is stretched or compressed by a mass, the spring develops a restoring force. Hooke's law gives the relationship of the force exerted by the spring when the spring is compressed or stretched a certain length: F ( t ) = − k x ( t ) , {\displaystyle F(t)=-kx(t),} where F is the force, k is the spring constant, and x is the ...
A mass suspended by a spring is the classical example of a harmonic oscillator A mass m attached to the end of a spring is a classic example of a harmonic oscillator . By pulling slightly on the mass and then releasing it, the system will be set in sinusoidal oscillating motion about the equilibrium position.
Notice =, a homogeneous rod oscillates as if it were a simple pendulum of two-thirds its length. A heavy simple pendulum: combination of a homogeneous rod of mass m r o d {\displaystyle m_{\mathrm {rod} }} and length ℓ {\displaystyle \ell } swinging from its end, and a bob m b o b {\displaystyle m_{\mathrm {bob} }} at the other end.
The spring-mass system illustrates some common features of oscillation, namely the existence of an equilibrium and the presence of a restoring force which grows stronger the further the system deviates from equilibrium. In the case of the spring-mass system, Hooke's law states that the restoring force of a spring is: =
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This point is located under the center of mass of the pendulum, at a distance which depends on the mass distribution of the pendulum. If most of the mass is concentrated in a relatively small bob compared to the pendulum length, the center of oscillation is close to the center of mass. [15]