Simple Harmonic Motion Frequency

The frequency of simple harmonic motion like a mass on a spring is determined by the mass m and the stiffness of the spring expressed in terms of a spring constant k ( see Hooke's Law):

Mass on Spring Resonance

A mass on a spring has a single resonant frequency determined by its spring constant k and the mass m. Using Hooke's law and neglecting damping and the mass of the spring, Newton's second law gives the equation of motion: 挂在弹簧上的质量具有由其弹簧常数k和质量m决定的单一共振频率。利用胡克定律并忽略阻尼和弹簧的质量，牛顿第二定律给出了运动方程： The solution to this differential equation is of the form: 这个微分方程的解形式为： which when substituted into the motion equation gives: 将其代入运动方程得到： Collecting terms gives B=mg/k, which is just the stretch of the spring by the weight, and the expression for the resonant vibrational frequency: 合并同类项得 B=mg/k，这正是弹簧因重量而产生的形变，且共振振动频率的表达式为： This kind of motion is called simple harmonic motion and the system a simple harmonic oscillator. 这种运动称为简谐运动，系统是一个简谐振子。

Mass on Spring: Motion Sequence

A mass on a spring will trace out a sinusoidal pattern as a function of time, as will any object vibrating in simple harmonic motion. One way to visualize this pattern is to walk in a straight line at constant speed while carriying the vibrating mass. Then the mass will trace out a sinusoidal path in space as well as time.

Show energy transformation involved in this motion

Energy in Mass on Spring

The simple harmonic motion of a mass on a spring is an example of an energy transformation between potential energy and kinetic energy. In the example below, it is assumed that 2 joules of work has been done to set the mass in motion.