Simple harmonic vibration refers to the motion of an object vibrating back and forth near its equilibrium position at a fixed frequency under the action of a restoring force. The formula can be expressed as:
x(t) = A * cos(ωt φ)
Where:
- x(t) is the object in time The displacement at t;
- A is the amplitude, indicating the maximum displacement of the object;
- ω is the angular frequency, and there is a relationship with the period T of vibration: ω = 2π / T;
- t is time;
- φ is the phase constant, which determines the starting phase of the vibration.
The characteristic of simple harmonic motion is that its displacement has a sinusoidal function relationship with time, which is periodic and symmetrical. Simple harmonic motion is widely used in physics, engineering and other scientific fields. Spring oscillators, pendulum clocks, etc. are all examples of simple harmonic motion.
Simple harmonic oscillation has the following characteristics
1. Periodicity
Simple harmonic oscillation vibrates with a fixed period T, that is, it repeats at equal time intervals Same movement. The period T is the time it takes for one vibration to occur.
2. Symmetry
The displacement-time curve of simple harmonic motion is a sine function or cosine function and has symmetry. When an object is in an equilibrium position, its displacement is zero; when it reaches its maximum displacement, its velocity is zero.
3. The restoring force is proportional to the displacement
The restoring force experienced by simple harmonic motion is proportional to the displacement of the object, and the direction of the restoring force is opposite to the direction of the displacement. This is consistent with Hooke's law, which is F = -kx, where F is the restoring force, k is the restoring force constant, and x is the displacement.
4. Maximum displacement and amplitude
The maximum displacement of simple harmonic motion is defined as the amplitude (A), which represents the distance of the object from the equilibrium position to the maximum displacement. The amplitude depends on the initial conditions and the energy of the system.
5. Angular frequency and angular velocity
The angular frequency (ω) of simple harmonic vibration is defined as the reciprocal of the vibration period, that is, ω = 2π / T. Angular velocity (ω) represents the rate of change of the phase angle that vibration passes through per unit time.
6. Conservation of energy
In simple harmonic motion, mechanical energy (kinetic energy and potential energy) is conserved. The kinetic energy of an object is greatest when it is at maximum displacement, and the potential energy is greatest when it is in an equilibrium position.
These characteristics together describe the basic characteristics of simple harmonic motion, making simple harmonic motion an important concept and model in physics and engineering.
Application of the simple harmonic vibration formula
1. Physics
The simple harmonic vibration formula is used to describe physical systems such as spring oscillators, simple harmonic pendulums, and sound waves. vibration. Their motion can be modeled and analyzed using simple harmonic oscillation formulas to study their characteristics such as frequency, amplitude, and phase.
2. Engineering
The simple harmonic vibration formula is also widely used in engineering. For example, for bridges, buildings or mechanical systems in structural engineering, the simple harmonic vibration formula can be used to study their vibration response when they are excited by external forces to evaluate the stability and safety of the structure.
3. Circuit Science
The simple harmonic vibration formula is applied to the analysis of AC circuits in circuit science. For example, in oscillation circuits, such as LC oscillation circuit, RC oscillation circuit and resonant circuit, the simple harmonic vibration formula can be used to describe the vibration behavior of voltage and current.
4. Optics
In the field of optics, the simple harmonic vibration formula can be used to describe the propagation and vibration of light waves. For example, by representing electromagnetic fields as simple harmonic vibrations, phenomena such as interference, diffraction, and polarization of light can be better understood.
5. Musicology
Sounds in music can also be described using the simple harmonic vibration formula. The tone and timbre produced by musical instruments can be analyzed and explained through the simple harmonic vibration formula.
Examples of the simple harmonic motion formula
Problem: A particle vibrates near the equilibrium position in a simple harmonic motion with an amplitude of 0.1 m and an angular frequency of 5 rad/s. Find:
a) The displacement function of the particle;
b) The displacement and velocity of the particle at t = 2 s.
Answer:
a) The general form of the displacement function is x(t) = A * cos(ωt φ), where A is the amplitude, ω is the angular frequency, and t is the time ,φ is the phase difference.
According to the information given in the question, the amplitude A = 0.1 m and the angular frequency ω = 5 rad/s. Therefore, the displacement function of the particle is x(t) = 0.1 * cos(5t φ).
b) When t = 2 s, by substituting the value of t, the displacement and velocity of the particle can be calculated.
Displacement: x(2) = 0.1 * cos(5 * 2 φ)
Velocity: v(2) = dx/dt = -0.1 * 5 * sin(5 * 2 φ)
Note: Due to the lack of specific initial velocity or phase information, specific displacement and velocity values ??cannot be obtained. Only their expressions can be obtained.