# Spring Mass system where springs also have mass

• sleventh
In summary, the conversation is about modeling a spring system with mass and understanding the potential energy and force system. The design involves vertical springs connected to two masses and is grounded on both ends. The suggested equation for potential energy is mgx+ 1/2k(x+δ)2 and the center of mass of the spring would move down the distance x.

#### sleventh

hello all,
this will be my third time posting this question, i thought perhaps posting in the homework section might have it be answered. Recently I have been attempting to model a spring system (see below) where the springs themselves will have mass. I should know how to do this but i accept defeat, how do you model the potential energy and force system on two masses connected to three springs? I have read about effective mass but i fail to fully understand. Thank you all for the celerity.
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In this design the spring system is vertical and grounded on both ends. Many thanks!

well the potential would just be mgx+ 1/2k(x+δ)2 if you plan to use Lagrange's equations to get your equations of motion.

The center of mass of the spring would move down the distance x as well as the mass m.

## 1. How does the mass of the spring affect the behavior of the spring-mass system?

The mass of the spring affects the natural frequency and amplitude of the oscillations in the spring-mass system. A heavier spring will have a lower natural frequency and a larger amplitude compared to a lighter spring. This is because the mass of the spring adds to the total mass of the system, changing the inertia and affecting the oscillatory motion.

## 2. Does the mass of the spring have an impact on the spring constant?

Yes, the mass of the spring does impact the spring constant. A heavier spring will have a lower spring constant, while a lighter spring will have a higher spring constant. This is because the spring constant is a measure of the stiffness of the spring and is inversely proportional to the mass of the spring. A heavier spring will be less stiff and thus have a lower spring constant.

## 3. What is the significance of considering the mass of the spring in a spring-mass system?

Considering the mass of the spring is important because it affects the accuracy of the model and the behavior of the system. Ignoring the mass of the spring can result in errors in predicting the natural frequency, amplitude, and other characteristics of the system. In real-world applications, springs do have mass and it is important to consider it in order to accurately model and understand the behavior of the system.

## 4. How can the mass of the spring be measured?

The mass of the spring can be measured using a scale or balance. The spring can be weighed before it is attached to the system, or the entire system can be weighed and the mass of the spring can be calculated by subtracting the mass of the other components. It is important to ensure that the spring is at rest and not undergoing any oscillations when measuring its mass.

## 5. Can the mass of the spring be changed or adjusted in a spring-mass system?

Yes, the mass of the spring can be changed or adjusted in a spring-mass system. This can be done by either replacing the spring with one of a different mass, or by adding or removing weight from the spring. Changing the mass of the spring will affect the behavior of the system, specifically the natural frequency and amplitude of the oscillations.