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Limitations of Homogenity

Paraser asked

“What are the limitations of homogenity?”

The question is acutually”What are the limitations of dimensional analysis?”

The dimensional analysis is based on the principle of homogenity of dimensions. The question is already discussed earlier  HERE

M – Theory, String Theory and more

Nijisha’s Question: There are various forces we see in nature, and among them, gravitational force is the weakest. If we think about it ,the entire earth is pulling on us and yet we can manage to pick thinks up. I’ve heard somewhere that we can explain this weakness of gravitational force with the help of extra dimensions ( Something related to M theory? ). Can you explain this theory ?

Ans:

It’s a little bit difficult to discuss in detail the M theory in detail as such at School level. However, I’ll try to give a brief idea about it.

M-theory is an extension of string theory in which 11 dimensions are identified. This is a theory currently in development

To quote from Wikipedia:

“In the 1980s, a new mathematical model of theoretical physics called string theory emerged. It showed how all the particles, and all of the forms of energy in the universe, could be constructed by hypothetical one-dimensional “strings,” infinitely small building-blocks that have only the dimension of length, but not height or width. Further, string theory suggested that the universe is made up of multiple dimensions. We are familiar with height, width, and length as three dimensional space, and time gives a total of four observable dimensions. However, string theories initially supported the possibility of ten dimensions—the remaining 6 of which we can’t detect directly. This was later increased to 11 dimensions based on various interpretations of the ten dimensional theory”

In M-theory may stand for membrane. (Some claim M for mother, matrix,mystery etc)

M-Theory brought all of the string theories together. It did this by asserting that strings are really 1-dimensional slices of a 2-dimensional membrane vibrating in 11-dimensional space.

Though M – Theory is developed mathematically, it is not proved physically. Scientists and Mathematicians take M-Theory as the only theory developed so far which can explain all the aspects of Universe (or multiverse) as a whole. But, I am sure, new theories will come and take over as we develop our understanding further.

 

Stephen Hawking’s New work The Grand Design talks about the Multiverse and the God’s intervention in the creation of Universe (which he says not necessary)

Ref:

(I fear that I’ve brought more unknowns in explaining the one unknown. However, it will be good if the discussion goes on.)

Difference between g and G

What is the difference between capital ‘G’ and small ‘g’ in the chapter gravitation?

 g  G
 Acceleration due to gravity  Universal constant of gravitation
 varies from place to place  A constant everywhere on earth
 vector quantity  A dimensional constant
 unit is ms-2  unit is Nm2kg-2

Dimensional Analysis – A solved example

Deduce the dimensional formula for workdone,pressure and density. [posted by boineelo ]

Answer:

workdone

We know that workdone = force x distance

Therefore [work done] =  [force] [distance]

but force = ma

[force]=[m][a]

[a]=[velocity]/[t]

[velocity]=[displacement]/[t]

So, putting all these together

[work done]=[m][distance][distance]/[time][time]

[work done]=ML2T-2

Similarly, you can work out the dimensional formula for pressure and density as

[pressure]=[force]/[area]=ML-1T-2

[density]=[mass]/[volume]=ML-3

Learn Physics Through Videos (Khan Academy)

One-dimensional motion

In this tutorial we begin to explore ideas of velocity and acceleration. We do exciting things like throw things off cliffs (far safer on paper than in real life) and see how high a ball will fly in the air.

Two-dimensional motion

You understand velocity and acceleration well in one-dimension. Now we can explore scenarios that are even more fun. With a little bit of trigonometry (you might want to review your basic trig, especially what sin and cos are), we can think about whether a baseball can clear the “green monster” at Fenway Park.

Forces and Newton’s laws of motion

This is the meat of much of classical physics. We think about what a force is and how Newton changed the world’s (and possibly your) view of how reality works.

Work and energy

“Energy” is a word that’s used a lot. Here, you’ll learn about how it’s one of the most useful concepts in physics. Along the way, we’ll talk about work, kinetic energy, potential energy, conservation of energy, and mechanical advantage.

Impacts and linear momentum

Momentum ties velocity and mass into one quantity. It might not be obvious why this is useful, but momentum has this cool property where the total amount of it never changes. This is called the conservation of momentum, and we can use it to analyze collisions and other interactions. Bam!

Moments, torque, and angular momentum

Everything you’ve learned about motion, forces, energy, and momentum can be reused to analyze rotating objects. There are some differences, though. Here, you’ll learn about rotational motion, moments, torque, and angular momentum.

Gravitation

Gravity is the force of attraction between masses. It’s the thing that pulls you down to earth. Here, you’ll learn precise meanings of the words mass and weight, and you’ll also learn how gravity affects falling near earth and orbits in space.

Oscillatory motion

Pendulums. Slinkies. You when you have to use the bathroom but it is occupied. These all go back and forth over and over and over again. This tutorial explores this type of motion.

Fluids

Atmospheric pressure is like an invisible friend who is always squeezing you with a big hug. Learn more about pressure, buoyant force, and flowing fluid so you can appreciate the sometimes invisible, but crucial, effect they have on us and the world around us.

Thermodynamics

Heat can be useful, but it can also be annoying. Understanding heat and the flow of heat allows us to build heat sinks that prevent our computers from overheating, build better engines, and prevent freeway overpasses from cracking.

Electric charge, electric force, and voltage

Electric forces hold together the atoms and molecules in your eyes which allow you to read this sentence. Take a moment and learn about the force that holds our bodies together.

Circuits

Circuits make computers, digital cameras, and video games possible. Circuits are driving an unprecedented rate of change in how we live. In this topic you’ll learn about the physics behind the electronic devices we use.

Magnetic forces and magnetic fields

The magnetic field of the Earth shields us from harmful radiation from the Sun. Magnetic fields also allow us to diagnose medical problems using an MRI. In this topic you’ll learn about the force and field that makes this possible.

Magnetic flux and Faraday’s law

Faraday’s law is how we get electrical power from most power plants and hydroelectric dams. Learn how magnetic flux allows us to turn the mechanical energy of falling water through a dam into electrical energy.

Mechanical waves and sound

Waves are responsible for basically every form of communication we use. Whether you’re talking out loud, texting on your phone, or waving to someone in a crowd there’s going to be a wave transmitting information. Learn about the basics of waves in this topic, then learn more about light waves in the topics below.

Light waves

Light can seem mysterious. What is light made out of? What causes color? How do 3D movies work? Learn about some of the mysterious properties of light in these tutorials.

Geometric optics

Light waves can be bent and reflected to form new and sometimes altered images. Understanding how light rays can be manipulated allows us to create better contact lenses, fiber optic cables, and high powered telescopes.

Special relativity (sneak peek)

Think you know about time and space? Think again. Einstein basically did a pile driver on all our brains when he came up with his theory of special relativity. Note: This topic is under construction. More videos and materials will be added soon.

Limitations of Dimensional Analysis

Ashmeeta Bhattarai asked:

“What are limitations of principle of homogeneity of dimensional analysis?”

The dimensional analysis has the following limitations

  1. It fails while using it to derive a relation among physical quantities, if there are more than 3 unknown variables on which a given physical quantity depends
  2. It does not tell whether a given Physical quantity is a scalar or a vector.
  3. It does not tell us the value of constants involved
  4. It does not always tell us the exact FORM of a relation
  5. It cannot be used for deriving logarithmic, trigonometric or exponential relations
  6. A dimensionally correct equation may not always be the correct relation. (Because there are more than one physical quantity having the same dimensions)

Know some more? Write them as comments to this post

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