# Help differentiating energy wrt time.

• caius
In summary: Thanks for your help!In summary, the mass is suspended in a system of two springs, and the equation to determine the total energy of the system is E = 1/2 mv^2 + k(x-l)^2 + 2k(l-x)^2 -mgx. The attempt at a solution is to substitute dx/dt in place of the velocity and differentiate. After making the corrections, the terms will be all forces, including the springs, and the answer will be m.a equated with the spring forces according to Hooke's law and gravity.
caius

## Homework Statement

I have a problem where I have a mass suspended in a system of springs. I need to differentiate the equation wrt time so I can can show equivalence with Newton's second law.

The mass and springs are vertically aligned so the motion is in one dimension. The actual problem has several springs, but for simplicity I am describing a system with just two. The equation below I think shows the total energy of the system.

## Homework Equations

E = 1/2 mv^2 + k(x-l)^2 + 2k(l-x)^2 -mgx

where m=mass, v= velocity, k= stiffness, x=current position and l=spring's natural length.

## The Attempt at a Solution

I think the way to approach it is to substitute dx/dt in place of the velocity, however I can't see what to do with the spring parts.

I seem to have some kind of mental block on this, and it's very frustrating. Any assistance on how to approach it would be gratefully received!

E = 1/2 mv^2 + k(x-l)^2 + 2k(l-x)^2 -mgx - don't think this is quite right

I am guessing that we have one spring above the mass and one below?

The terms for elastic spring energy are based on 1/2ke^2. Where e is spring extension.

Isn't the extension of the springs l+x and l-x?

Are both springs of the same stiffness constant k? Perhaps there is a typo here?

Make these corrections and multiply out brackets before differentiating.

Remember
$$\frac{d}{dt}v^2=2v\frac{dv}{dt}=2\frac{dx}{dt}a$$
Where a is acceleration. So the kinetic energy term after differentiating will have ma in it, which is starting to look like N2L. It will also have dx/dt in it. All other terms of your equation will either go to zero because they are independent of t or be a multiple of dx/dt. So dx/dt will cancel throughout. This will leave terms that are all forces.

apelling said:
Isn't the extension of the springs l+x and l-x?
I don't see how it could be l+x. l-x and x-l make sense if the springs are in series with the endpoints fixed 2l apart, and x being the position of the join. But then, the expression simplifies to 3k(l-x)2.

Thanks for your suggestions. I think I have it now.

The lengths were the correct way round, but I didn't make it quite clear in the question I asked. The distance between the two fixtures is known, so the deformation can be expressed in relation to that.

The answer I got did end up as m.a equated with the spring forces according to Hooke's law and gravity which is what I needed.

It's amazing the difference a nights sleep makes.

haruspex said:
I don't see how it could be l+x. l-x and x-l make sense if the springs are in series with the endpoints fixed 2l apart, and x being the position of the join. But then, the expression simplifies to 3k(l-x)2.

I was thinking L was the extension of the springs when at equilibrium and x was the displacement from equilibrium. Anyhow it does not matter much since the constants drop out during differentiation.

## 1. What is the concept of energy differentiation with respect to time?

The concept of energy differentiation with respect to time is a mathematical process used to determine how energy changes over a specific period of time. It involves calculating the rate of change of energy with respect to time, also known as the energy derivative.

## 2. How is energy differentiation different from regular differentiation?

Energy differentiation is different from regular differentiation in that it specifically focuses on changes in energy over time, while regular differentiation can be applied to any variable. In energy differentiation, the units of measurement are also specific to energy, such as joules per second.

## 3. What is the practical application of energy differentiation?

The practical application of energy differentiation is in physics and engineering, where it is used to analyze and understand systems that involve energy, such as motion, heat transfer, and electrical circuits. It can also be used in economics to analyze changes in energy consumption over time.

## 4. Can energy differentiation be negative?

Yes, energy differentiation can be negative. If the rate of change of energy with respect to time is negative, it means that the energy is decreasing over time. This could happen, for example, when a moving object is slowing down and losing kinetic energy.

## 5. Is energy differentiation a one-time calculation or an ongoing process?

Energy differentiation is an ongoing process, as energy can continue to change over time. To accurately analyze a system, multiple energy differentiations may need to be performed at different points in time to capture changes in energy over the entire duration of the process.

Replies
24
Views
2K
Replies
29
Views
2K
Replies
17
Views
699
Replies
5
Views
1K
Replies
3
Views
2K
Replies
2
Views
1K
Replies
6
Views
398
Replies
8
Views
716
Replies
2
Views
905
Replies
7
Views
2K