# Equations of the harmonic oscillator

• KodRoute
In summary: For figure (c), focus on the small right-triangle that has ##\vec v_t## as its hypotenuse. What linear speed (m/sec) does that correspond to? (This is the magnitude of the vector ##\vec v_t##.)
KodRoute
Hello, my book explains detailed the proofs of these three formulas:

y = Asin(ωt + φo)
v = ωAcos(ωt + φo)
a = -ω²Asin(ωt + φo)

Where a is acceleration, v is velocity, ω is angular velocity, A is amplitude.

The book uses the following figures:
Figure a) --> http://tinypic.com/view.php?pic=30m8f9j&s=8
Figure b) --> http://tinypic.com/view.php?pic=2hrejde&s=8
Figure c) --> http://tinypic.com/view.php?pic=a0f9te&s=8

In the first case a) what I'm seeing is that the sine in the triangle whose hypotenuse is R and opposite side vector y is (vector y)/(vector R), in this case R is the amplitude.

The rest b) and c) are just complicated, I don't understand them. Anyone out there to help me here?
Thank you!

Edit: Sorry for posting this here, it's supposed to go to homework help, my fault.

Conceptual questions that arise from not understanding your textbook or lecturer's notes, are fine here. The homework-help forums are for getting help with solving specific exercises.

The simplest way to derive the formulas for v and a is by using calculus. v is the derivative of y with respect to t, and a is the derivative of v with respect to t. I suppose you haven't had calculus yet?

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jtbell said:
Conceptual questions that arise from not understanding your textbook or lecturer's notes, are fine here. The homework-help forums are for getting help with solving specific exercises.

The simplest way to derive the formulas for v and a is by using calculus. v is the derivative of y with respect to t, and a is the derivative of v with respect to t. I suppose you haven't had calculus yet?
Hi! Thank you for replying, no I haven't done calculus yet. I thought these formulas came from looking at the geometry of these vectors.

For figure (b), focus on the small right-triangle that has ##\vec v_t## as its hypotenuse. Which trig function (sin or cos) is associated with the y-component of ##\vec v_t##, and is that component + or -?

To get the amplitude (maximum value) of v, remember that the point moves around the circle at constant angular speed ω (radians/sec). What linear speed (m/sec) does that correspond to? (This is the magnitude of the vector ##\vec v_t##.)

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No problem, I'm happy to help with understanding the equations of the harmonic oscillator. The equations you listed are commonly used to describe the motion of a harmonic oscillator, which is a system that oscillates or vibrates around an equilibrium point. This can be seen in Figure a) where the object is moving back and forth between the two extremes, represented by the amplitude A.

To understand the equations, it may be helpful to think about the physical meaning of each term. The variable y represents the displacement of the object from its equilibrium point at any given time t. This is why it is multiplied by the sine function, which describes the oscillating motion. The amplitude A represents the maximum displacement of the object, while ω is the angular velocity, or how quickly the object is oscillating. The φo term represents the initial phase of the oscillation, which can be thought of as the starting position of the object.

The next two equations describe the velocity and acceleration of the object. As you can see, the velocity equation includes a cosine function, which is the derivative of sine and represents the rate of change of the displacement. This is why it is multiplied by ω, as it represents the speed at which the object is oscillating. The acceleration equation is the derivative of the velocity equation, which is why it includes the ω² term. This term represents the change in the velocity over time, and the negative sign indicates that the acceleration is in the opposite direction of the displacement.

In Figures b) and c), the book is likely showing how these equations can be derived using calculus and the principles of motion. This is why they may seem complicated, but they provide a more in-depth understanding of the equations and their physical meaning.

I hope this helps clarify the equations for you. If you have any further questions, please don't hesitate to ask. As a scientist, it's important to have a solid understanding of the equations we use in our work, so I'm glad you're seeking clarification.

## 1. What is the equation for a harmonic oscillator?

The equation for a harmonic oscillator is m¨ + kx = 0, where m is the mass of the oscillator, k is the spring constant, and x is the displacement from equilibrium.

## 2. What is the physical significance of the harmonic oscillator equation?

The harmonic oscillator equation describes the motion of a system that is subject to a restoring force that is proportional to the displacement from equilibrium. This type of motion is commonly seen in simple systems such as a mass attached to a spring.

## 3. How is the period of a harmonic oscillator related to the mass and spring constant?

The period of a harmonic oscillator is T = 2π√(m/k), where T is the period, m is the mass, and k is the spring constant. This means that the period is inversely proportional to the square root of the mass and directly proportional to the square root of the spring constant.

## 4. What is the difference between a simple harmonic oscillator and a damped harmonic oscillator?

A simple harmonic oscillator is an idealized system with no external forces acting on it, while a damped harmonic oscillator experiences external forces that gradually decrease its amplitude over time. This can be due to friction or other dissipative forces.

## 5. How does the amplitude of a harmonic oscillator change with increasing energy?

The amplitude of a harmonic oscillator increases with increasing energy. This is because higher energy levels correspond to larger displacements from equilibrium, resulting in a larger amplitude of motion. However, this relationship is not linear, as the amplitude also depends on the mass and spring constant of the system.

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