# In what cases does dimensional analysis fail?

• I
• VVS2000
In summary, the conversation revolved around the use of dimensional analysis in cases where the nature of the opposing quantity is unknown, such as in the case of damping in oscillations. The expert suggested that dimensional analysis can still be used in such cases and provided a helpful link to further explain its use in differential equations. The expert also asked for the dimensions of the terms in the initial differential equation to show how it can help define the dimensions of λ.
VVS2000
TL;DR Summary
In what cases does dimensional analysis fails? And also is there like any preferred situation or an 'ideal' situation in order to use dimensional analysis? Thanks in advance!
It possible a diagram would be really really helpful

Is this a schoolwork question?

No no Like they were just going through dimensional analysis in our class and I just thought well obviously this won't work for evert case so I just wanted to know what are those cases.

VVS2000 said:
Summary: In what cases does dimensional analysis fails?
Well, a number of quantities are dimensionless, but I'm not sure that represents a failure. Can you think of some dimensionless constants or quantities that are used commonly?

Also, these two PF Insights articles would probably be good background reading for you to help understand dimensional analysis better...

https://www.physicsforums.com/insights/learn-the-basics-of-dimensional-analysis/
https://www.physicsforums.com/insights/make-units-work/

VVS2000
VVS2000 said:
I just thought well obviously this won't work for evert case so I just wanted to know what are those cases.

That's like asking when addition doesn't work - when it's the wrong tool for the job. Maybe you need to subtract or to multiply.

Dimensional analysis cannot tell you if the circumference of a circle is 6r, 2πr or 7r. It can tell you it is not 1/r or r2.

anorlunda
berkeman
That's like asking when addition doesn't work - when it's the wrong tool for the job. Maybe you need to subtract or to multiply.

Dimensional analysis cannot tell you if the circumference of a circle is 6r, 2πr or 7r. It can tell you it is not 1/r or r2.
No no not like that. What if sometimes we don't know the nature of the opposing quantity. How are you going to proceed then?

VVS2000 said:
What if sometimes we don't know the nature of the opposing quantity.

I don't understand.

I don't understand.
Like in damping of harmonic oscillations.

VVS2000 said:
Like in damping of harmonic oscillations.
What do the overall dimensions of the quantity in the exponent need to be? So what does that tell you about the dimensions of λ ?

http://hyperphysics.phy-astr.gsu.edu/hbase/oscda.html

I still don't understand. Perhaps you could use complete sentences? Maybe even paragraphs?

nasu and SammyS
I still don't understand. Perhaps you could use complete sentences? Maybe even paragraphs?
For example damping in oscillations. What if we don't know the nature of damping? How do you find the equations of motion?

VVS2000 said:
How do you find the equations of motion?

What does that have to do with dimensional analysis?

VVS2000 said:
For example damping in oscillations. What if we don't know the nature of damping? How do you find the equations of motion?

What does that have to do with dimensional analysis?
I am asking using dimensional analysis how do you find the eqaution of motion then

berkeman said:
Sorry I am new here. Post 10 where?

No worries. The post number in each thread is shown in the upper right of each post. This post of mine will likely be #17, since I'm replying to your post #16. So just scroll up the thread to look for my reply in post #10. It was a post showing some equations from Hyperphysics to help you see the typical differential equation and solution for a damped harmonic oscillator.

berkeman said:
No worries. The post number in each thread is shown in the upper right of each post. This post of mine will likely be #17, since I'm replying to your post #16. So just scroll up the thread to look for my reply in post #10. It was a post showing some equations from Hyperphysics to help you see the typical differential equation and solution for a damped harmonic oscillator.

VVS2000 said:
So I cannot use dimensional analysis in such cases?

berkeman said:
No worries. The post number in each thread is shown in the upper right of each post. This post of mine will likely be #17, since I'm replying to your post #16. So just scroll up the thread to look for my reply in post #10. It was a post showing some equations from Hyperphysics to help you see the typical differential equation and solution for a damped harmonic oscillator.
So I cannot use dimensional analysis in such cases?

VVS2000 said:
So I cannot use dimensional analysis in such cases?
Like in the Hyperphysics equations? Sure. Can you show the dimensions for each of the terms in the initial differential equation? And then show how that helps to define the dimensions of λ in the end?

berkeman said:
Like in the Hyperphysics equations? Sure. Can you show the dimensions for each of the terms in the initial differential equation? And then show how that helps to define the dimensions of λ in the end?
I think I can. I can show the complimentary function for x. Then I can By eliminating the constants by plugging in the initial conditions, we can find the dimensions of lambda

VVS2000 said:
I think I can. I can show the complimentary function for x. Then I can By eliminating the constants by plugging in the initial conditions, we can find the dimensions of lambda
Okay, but what are the dimensions (or units) for each of the terms in these equations?

VVS2000 said:
I think I can. I can show the complimentary function for x. Then I can By eliminating the constants by plugging in the initial conditions, we can find the dimensions of lambda
The dimensions of c are...?
The dimensions of m are...?
The dimensions of ##\lambda## are...?

The dimensiona of lambda is (time)^-1, time inverse, basically frequency. .M is mass. That is given. I don't or I am not quite sure on c

VVS2000 said:
The dimensiona of lambda is (time)^-1, time inverse, basically frequency. .M is mass. That is given. I don't or I am not quite sure on c
The starting equation contains forces. What are the units of force in mks? Each term has to have those same units, right? That helps you figure out the units for the other terms that introduced in the later equations. And yes, since λt is in the exponent, and since the exponent needs to be dimensionless, the units of λ need to be 1/s.

VVS2000 said:
In what cases does dimensional analysis fails?
Not sure what 'fails' means here, but dimensional homogeneity doesn't guarantee that your equation is physically sensible:

https://en.wikipedia.org/wiki/Dimensional_analysis#Dimensional_homogeneity
Even when two physical quantities have identical dimensions, it may nevertheless be meaningless to compare or add them. For example, although torque and energy share the dimension L2MT−2, they are fundamentally different physical quantities.

berkeman
berkeman said:
The starting equation contains forces. What are the units of force in mks? Each term has to have those same units, right? That helps you figure out the units for the other terms that introduced in the later equations. And yes, since λt is in the exponent, and since the exponent needs to be dimensionless, the units of λ need to be 1/s.
Yeah ok so c has units of M(T)^-1

A.T. said:
Not sure what 'fails' means here, but dimensional homogeneity doesn't guarantee that your equation is physically sensible:

https://en.wikipedia.org/wiki/Dimensional_analysis#Dimensional_homogeneity
Even when two physical quantities have identical dimensions, it may nevertheless be meaningless to compare or add them. For example, although torque and energy share the dimension L2MT−2, they are fundamentally different physical quantities.
Yeah sorry. I think I did'nt frame my question in the proper way. So the math part of the equation can be checked out but it will not be wise to try and infer the physics aspect of it

VVS2000 said:
The dimensiona of lambda is (time)^-1, time inverse, basically frequency. .M is mass. That is given. I don't or I am not quite sure on c
That’s correct.

## 1. What is dimensional analysis?

Dimensional analysis is a mathematical technique used to analyze and solve problems involving physical quantities. It involves examining the dimensions (such as length, time, mass, etc.) of the quantities involved in a problem to determine the relationships between them.

## 2. How does dimensional analysis help in problem-solving?

Dimensional analysis helps in problem-solving by providing a systematic approach to understanding the relationships between physical quantities. It allows scientists to check the correctness of equations and to convert between different units of measurement.

## 3. In what cases does dimensional analysis fail?

Dimensional analysis may fail in cases where there are non-linear relationships between physical quantities or when there are multiple variables involved. It also cannot account for dimensionless quantities, such as ratios or percentages.

## 4. Can dimensional analysis be used in all branches of science?

Yes, dimensional analysis can be used in all branches of science, including physics, chemistry, biology, and engineering. It is a fundamental tool for understanding and solving problems involving physical quantities.

## 5. How can dimensional analysis be used in real-life situations?

Dimensional analysis can be used in real-life situations to convert between different units of measurement, to check the validity of equations, and to solve problems involving physical quantities. For example, it can be used to convert units of length when traveling to a different country or to calculate the amount of medication needed based on a patient's weight.

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