15.4
简谐振动的能量
要确定简谐振子的能量，须考虑它在简谐振动期间可能具有的所有形式的能量。 根据胡克定律，简谐振子中的弦压缩/拉伸期间存储的能量是势能。 由于简谐振子没有耗散力，因此它也具有动能。 在存在保守力的情况下，两种能量可以在振荡过程中相互转换，但总能量保持恒定。 简谐振子的总能量等于势能和动能的总和，并与振幅…
考虑一个质量为 m 的物块连接在劲度系数为 k 的弹簧上，置于无摩擦表面上并处于平衡状态。
当弹簧发生形变时，弹簧力在从初始位置到最终位置的位移过程中所做的功，等于弹簧相对于其形变量所储存的势能。这种能量称为弹性势能。
当释放该物块时，它将进行简谐运动。其往复运动所需的能量称为平动动能，与物块速度的平方及其质量成正比。
在振荡过程中，两种能量持续相互转换，并以正弦波形表示。当位移最大时，弹性势能达到最大值；而在平衡位置时，平动动能达到最大值。
在其他位置，该物体具有不同的动能和势能值，二者之和等于系统的总能量。
因此，系统中的总能量保持恒定且守恒，在平动动能和弹性势能之间相互转换。
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Q1: What is elastic potential energy in a spring-mass system?
Elastic potential energy is the energy stored in a spring when it is compressed or stretched from equilibrium. According to Hooke's Law, the work done by the spring force during displacement equals the potential energy stored relative to that displacement. This energy is maximum when the block reaches maximum displacement and is directly related to how far the spring is stretched or compressed.
Q2: How does translational kinetic energy change during simple harmonic motion?
Translational kinetic energy is the energy of motion and is directly proportional to both the mass and the square of the velocity. During oscillation, kinetic energy is maximum at the equilibrium position where velocity is highest, and zero at maximum displacement where the block momentarily stops. As the block moves, kinetic energy continuously converts to and from potential energy.
Q3: Why does total energy remain constant in a simple harmonic oscillator?
Total energy remains constant because the system has no dissipative forces like friction. The sum of elastic potential energy and translational kinetic energy stays the same throughout oscillation. Energy continuously interconverts between these two forms—when potential energy decreases, kinetic energy increases proportionally, and vice versa, maintaining constant total energy.
Q4: What factors determine maximum velocity in simple harmonic motion?
Maximum velocity depends on three factors: amplitude, force constant, and mass. Maximum velocity is proportional to amplitude and the square root of the force constant, but inversely proportional to the square root of mass. Stiffer springs produce greater maximum velocities for the same amplitude, while heavier objects move more slowly at maximum velocity.
Q5: How are potential and kinetic energy distributed at different positions during oscillation?
At maximum displacement, elastic potential energy is maximum and kinetic energy is zero. At equilibrium position, kinetic energy is maximum and potential energy is zero. At intermediate positions, both energies have intermediate values that sum to the constant total energy. This energy distribution creates the sinusoidal waveforms characteristic of simple harmonic motion.
Q6: How is total energy related to amplitude in a simple harmonic oscillator?
Total energy in a simple harmonic oscillator is proportional to the square of the amplitude. This means doubling the amplitude quadruples the total energy. Since total energy equals the sum of potential and kinetic energy, and this sum remains constant throughout oscillation, the amplitude directly determines how much total energy the system possesses.
Q7: Why is kinetic energy maximum at the equilibrium position?
At equilibrium, the spring exerts no restoring force, so all the system's energy exists as kinetic energy. The block passes through equilibrium with maximum velocity because all elastic potential energy has been converted to motion energy. Beyond equilibrium, the spring begins pulling the block back, converting kinetic energy back into potential energy.