Differential equations - harmonic motion

In summary, a 5 kilogram mass causes a spring to stretch centimeters in equilibrium. When it oscillates vertically in a fluid, it experiences a frictional force times its speed. If it starts from rest at an extension below equilibrium, the subsequent motion can be described by the equation md^2x/dt^2 = mg - B - bdx/dt - kx, where m is the mass, g is the gravitational constant, B is the buoyant force, b is a constant, k is the spring constant, and x is the displacement from equilibrium. This is classified as damped harmonic oscillation.
  • #1
sugarplum31
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Homework Statement



A mass of 5 kilograms dangles from a spring, stretching the spring centimeters when in equilibrium.
The mass oscillates vertically in a fluid, with a frictional force times its speed.
If initially it starts from rest at an extension below the equilibrium point of centimeters, describe the subsequent motion, with a plot of the extension against time and with a phase space plot.


Homework Equations



I looked through my book, but my first problem is that I can't figure out if it is Simple Harmonic Motion, or Forced Harmonic Motion. I don't see any specific equations for this that involves all of the variables I have.

The Attempt at a Solution

 
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  • #2
Write the eqn of motion first. Take the downward direction as positive.

The weight mg acts downward, the buoyant force B acts upward, the frictional force bv acts opp to the velo, where b is a constant, and k is the spring constant, x is the displacement from the equilibrium posn.

ma = mg - B - bv - kx =>
md^2x/dt^2 = mg - B - bdx/dt - kx.

Sub in the values of the constants. This is damped harmonic oscillation.
 

Related to Differential equations - harmonic motion

1. What is a differential equation?

A differential equation is a mathematical equation that relates a function to its derivatives. In other words, it describes how the rate of change of a quantity is related to the quantity itself.

2. What is harmonic motion?

Harmonic motion is a type of periodic motion in which a system or object repeats its motion in a regular pattern over time. It follows the laws of simple harmonic motion, which states that the restoring force on the system or object is directly proportional to the displacement from its equilibrium position.

3. How are differential equations used to model harmonic motion?

Differential equations are used to model harmonic motion by describing the relationship between the position, velocity, and acceleration of an object undergoing harmonic motion. This can be represented by a second-order differential equation, such as the equation for simple harmonic motion, which is d²x/dt² = -ω²x, where ω is the angular frequency.

4. What is the significance of the natural frequency in harmonic motion?

The natural frequency in harmonic motion refers to the frequency at which the system or object will oscillate without any external forces acting on it. This frequency is determined by the physical properties of the system, such as its mass and stiffness, and can be calculated using the equation ω = √(k/m), where k is the spring constant and m is the mass.

5. How are differential equations used to solve problems related to harmonic motion?

Differential equations are used to solve problems related to harmonic motion by providing a mathematical model that can be used to predict the behavior of a system or object. By solving the differential equations, we can determine the position, velocity, and acceleration of the object at any given time, which can help us understand and analyze its motion.

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