Vibrations and differential equations.

In summary, the conversation discusses the problem of finding the differential equation for the motion of a spring with natural frequency 1/2 and an attached weight of 32lb, assuming no external force. The equation is given by 32y" + 8y = ? and the value of k is found to be 8. The question also addresses the need for finding the value of F_0 and ω, with the given information of natural frequency. It is mentioned that the natural frequency, f, can be related to the angular frequency, ω.
  • #1
benoconnell22
32
0

Homework Statement



Suppose the motion of a spring has natural frequency 1/2 and is undamped. If the weight attached is 32lb, write a differential equation describing the motion.

Homework Equations



my''+ky=F_ocosωt

32y"+8y=?

ω_o= (k/m)^.5

The Attempt at a Solution



→ .5=(k/32)^.5 → k=8

gamma=0

32y"+8y=?My problem is that I don't know how to find F_o or ω. Is this question referring to that of a no external force case? Or am I missing some equation that will help me find this?
 
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  • #2
The question is implying that there is no external forcing to F0 = 0. As for ω, they gave you the natural frequency f. How does this relate to the angular frequency ω?
 

1. What are vibrations and how are they related to differential equations?

Vibrations are a type of cyclic motion that occurs in many physical systems, such as a swinging pendulum or a vibrating guitar string. These motions can be described using differential equations, which are mathematical equations that relate the rate of change of a quantity to its current value. In the case of vibrations, differential equations help us understand how the position, velocity, and acceleration of an object change over time.

2. Why do we use differential equations to study vibrations?

Differential equations are useful for studying vibrations because they allow us to model complex systems and make predictions about their behavior. In many cases, it is not possible to solve these equations analytically, so numerical methods are used to approximate solutions. This allows us to understand and analyze vibrations in a wide range of systems, from mechanical structures to electrical circuits.

3. What are some real-world applications of vibrations and differential equations?

Vibrations and differential equations have many practical applications in engineering, physics, and other fields. For example, understanding the vibrations of bridges and buildings can help engineers design structures that can withstand earthquakes. In the field of acoustics, differential equations can be used to model the vibrations of musical instruments and design speakers and other audio equipment. In addition, differential equations are used in areas such as electronics, signal processing, and control systems.

4. How do initial conditions and boundary conditions affect the solutions of differential equations for vibrations?

Initial conditions and boundary conditions play a crucial role in determining the solutions of differential equations for vibrations. Initial conditions specify the values of position, velocity, and acceleration at a specific time, while boundary conditions define the behavior of the system at its boundaries. These conditions are essential for obtaining unique solutions to differential equations and can greatly affect the behavior of vibrating systems.

5. Are there any limitations to using differential equations to study vibrations?

While differential equations are a powerful tool for understanding vibrations, there are some limitations to their use. In some cases, it may not be possible to accurately model a system using differential equations, and other methods such as numerical simulations may be needed. Additionally, differential equations can become very complex for highly nonlinear systems, making it challenging to find solutions analytically. However, with advancements in computing technology, these limitations can often be overcome.

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