Dimensional Analysis as a test of equation plausibility

MartinT · July 25, 2016, 6:44pm

[Martin Taylor 2016.07.25.14.43]

I just realized I sent this under the wrong subject header. Sorry

about that.

···

[Martin Taylor 2016.07.25.13.13]
 Maybe the following will be self-evident when it is pointed out,
but it often seems to be ignored. If, say, an equation says that a
mass is equal to a distance, or a voltage is equal to a velocity,
there’s something wrong with the equation. Yet we often see
equations in which similarly impossible equalities are asserted.
So I thought it might be useful to offer a quick guide to
dimensional analysis.
 We start with a set of basic dimensions. They are basic in the
sense than none can be derived from operations on the others. One
possible set is distance, time, and mass, which we can symbolize
as L, T, and M, respectively. If these are taken as the basic
dimensions, then Force cannot be basic, since F=Ma, and
acceleration is based on distance and time There are other basic
dimensions such as electromagnetic ones, but L, T, and M will do
for my purpose in this message.
 It is impossible to add a mass to a time and get anything
sensible, or to add a distance to an area, an area being one
distance multiplied by another. Taking X and Y to be different
dimensions (basic or compound), we have the following rules.
 X + X = X (adding a distance to a distance yields a distance;
adding an area to an area yields an area)
 X * X = X2 (multiplying a distance by a distance yields
an area; multiplying X by itself n times yields Xⁿ.)
 X / X = 0 (dividing a distance by a distance give a dimensionless
quantity, just a number)
 X+Y = BAD (you can't add a distance to a mass)

 X * Y = XY (you can multiply a distance by a mass)

 X / Y = XY-1 (you can divide a distance by a time; the
result is called speed or velocity)
 An equation:

 X = X is plausible

 X = Y is not plausible



 Differentiation and integration.



 dX/dY = XY-1 . (a differential is of the same dimension
as its base quantity)
 d2X/dy2 = XY-1Y-1 = XY-2.

 Integral(X dY) = XY (the integral can be seen as the area under a
graph of X in terms of Y, and dY has the dimension of Y)
 Logs and exponentiation



 log(X) = X (the logarithm of a distance is a distance)

 exp(X) = X (because exp(log(v)) = v)



 Any compound such as XaYb can be substituted
for X and Y above, where X and Y are different compounds in any
one equation.
 The addition rules above work for subtraction as well as addition.



 Some examples:



 An angle measured in radians can be described by the ratio of the
arc of a circle to the radius of the circle, so it has dimensions
L/L = 0. It is a pure number. So is an angle measured in degrees
(but its scale is different when you use the equation with
measured variables). So is the rate at which a clock gains or
loses (seconds per day has dimension T/T = 0).
 A velocity is distance per unit time, or L/T = LT-1.

 An angular velocity is a number per unit time of T-1.



 The equation for the radius of a circle is R = sqrt(x2
 + y2) which dimensionally is (L2 + L2)-2
 = (L2)-2 = L. The equation is plausible.



 Non-integer exponents are allowed.

 >X'*Y''-X.''*Y.dot|1/3
 * R1/3 is (LT-1*LT-2 - LT-2*LT-1)1/3*L1/3
 =(L2T-3* - L2T-3)1/3*L1/3
 = (L2T-3)1/3*L1/3 
= L/T which is a linear velocity.
 dθ/ds = L-1 (angular change per unit path length, if θ
is an angle and s is a distance such as a portion of a curve)
 dθ/dt = T-1 (angular velocity, angular change per unit
time).
 d2s/dt2 (where s is a distance) yields LT-2 ,
so force has dimension MLT^-2 because F=ma.
 And so forth.



 Anyone can use dimensional analysis to check the plausibility of
their equations. You don’t have to be a mathematician. If the
dimensions on opposite sides of an equal sign differ, the equation
is wrong. It could be wrong even if the dimensions are the same,
but the plausibility check is often a good place to start. You can
also often use dimensional analysis to check whether you are
starting your problem solution with a plausible set of variables.
It’s a quick and easy way to find some easily made errors.
 Martin

rsmarken · July 25, 2016, 7:12pm

[From Rick Marken (2016.07.25.1212)]

···

Martin Taylor (2016.07.25.14.43)–

MT: I just realized I sent this under the wrong subject header. Sorry

about that.

RM: Is that because it’s not relevant to the discussion about the power curve?

  MT: If, say, an equation says that a
mass is equal to a distance, or a voltage is equal to a velocity,
there’s something wrong with the equation.

RM: What about that one that says mass is equal to mass times a constant?

RM: If this is supposed to be relevant to the power law could you explain how?

Best

Rick

Yet we often see
equations in which similarly impossible equalities are asserted.
So I thought it might be useful to offer a quick guide to
dimensional analysis.
 We start with a set of basic dimensions. They are basic in the
sense than none can be derived from operations on the others. One
possible set is distance, time, and mass, which we can symbolize
as L, T, and M, respectively. If these are taken as the basic
dimensions, then Force cannot be basic, since F=Ma, and
acceleration is based on distance and time There are other basic
dimensions such as electromagnetic ones, but L, T, and M will do
for my purpose in this message.
 It is impossible to add a mass to a time and get anything
sensible, or to add a distance to an area, an area being one
distance multiplied by another. Taking X and Y to be different
dimensions (basic or compound), we have the following rules.
 X + X = X (adding a distance to a distance yields a distance;
adding an area to an area yields an area)
 X * X = X2 (multiplying a distance by a distance yields
an area; multiplying X by itself n times yields Xⁿ.)
 X / X = 0 (dividing a distance by a distance give a dimensionless
quantity, just a number)
 X+Y = BAD (you can't add a distance to a mass)

 X * Y = XY (you can multiply a distance by a mass)

 X / Y = XY-1 (you can divide a distance by a time; the
result is called speed or velocity)
 An equation:

 X = X is plausible

 X = Y is not plausible



 Differentiation and integration.



 dX/dY = XY-1 . (a differential is of the same dimension
as its base quantity)
 d2X/dy2 = XY-1Y-1 = XY-2.

 Integral(X dY) = XY (the integral can be seen as the area under a
graph of X in terms of Y, and dY has the dimension of Y)
 Logs and exponentiation



 log(X) = X (the logarithm of a distance is a distance)

 exp(X) = X (because exp(log(v)) = v)



 Any compound such as XaYb can be substituted
for X and Y above, where X and Y are different compounds in any
one equation.
 The addition rules above work for subtraction as well as addition.



 Some examples:



 An angle measured in radians can be described by the ratio of the
arc of a circle to the radius of the circle, so it has dimensions
L/L = 0. It is a pure number. So is an angle measured in degrees
(but its scale is different when you use the equation with
measured variables). So is the rate at which a clock gains or
loses (seconds per day has dimension T/T = 0).
 A velocity is distance per unit time, or L/T = LT-1.

 An angular velocity is a number per unit time of T-1.



 The equation for the radius of a circle is R = sqrt(x2
 + y2) which dimensionally is (L2 + L2)-2
 = (L2)-2 = L. The equation is plausible.



 Non-integer exponents are allowed.

 |X'*Y''-X.''*Y.dot|1/3
 * R1/3 is (LT-1*LT-2 - LT-2*LT-1)1/3*L1/3
 =(L2T-3* - L2T-3)1/3*L1/3
 = (L2T-3)1/3*L1/3 
= L/T which is a linear velocity.
 dθ/ds = L-1 (angular change per unit path length, if θ
is an angle and s is a distance such as a portion of a curve)
 dθ/dt = T-1 (angular velocity, angular change per unit
time).
 d2s/dt2 (where s is a distance) yields LT-2 ,
so force has dimension MLT^-2 because F=ma.
 And so forth.



 Anyone can use dimensional analysis to check the plausibility of
their equations. You don’t have to be a mathematician. If the
dimensions on opposite sides of an equal sign differ, the equation
is wrong. It could be wrong even if the dimensions are the same,
but the plausibility check is often a good place to start. You can
also often use dimensional analysis to check whether you are
starting your problem solution with a plausible set of variables.
It’s a quick and easy way to find some easily made errors.
 Martin

–

Richard S. Marken

“The childhood of the human race is far from over. We
have a long way to go before most people will understand that what they do for
others is just as important to their well-being as what they do for
themselves.” – William T. Powers

rsmarken · July 25, 2016, 7:15pm

[From, Rick Marken (2016.07.25.1215)]

RM Fly larva brain attacks again!

MT: If, say, an equation says that a mass is equal to a distance, or a voltage is equal to a velocity, there's something wrong with the equation.

RM: What about that one that says mass is equal to mass times a constant?

RM: Of course, I means the one that says energy is equal to mass times a constant.
Best
Rick
>

RM: If this is supposed to be relevant to the power law could you explain how?
Best
Rick

Yet we often see equations in which similarly impossible equalities are asserted. So I thought it might be useful to offer a quick guide to dimensional analysis.

We start with a set of basic dimensions. They are basic in the sense than none can be derived from operations on the others. One possible set is distance, time, and mass, which we can symbolize as L, T, and M, respectively. If these are taken as the basic dimensions, then Force cannot be basic, since F=Ma, and acceleration is based on distance and time There are other basic dimensions such as electromagnetic ones, but L, T, and M will do for my purpose in this message.

It is impossible to add a mass to a time and get anything sensible, or to add a distance to an area, an area being one distance multiplied by another. Taking X and Y to be different dimensions (basic or compound), we have the following rules.

X + X = X (adding a distance to a distance yields a distance; adding an area to an area yields an area)
X * X = X2 (multiplying a distance by a distance yields an area; multiplying X by itself n times yields Xn.)
X / X = 0 (dividing a distance by a distance give a dimensionless quantity, just a number)

X+Y = BAD (you can't add a distance to a mass)
X * Y = XY (you can multiply a distance by a mass)
X / Y = XY-1 (you can divide a distance by a time; the result is called speed or velocity)

An equation:
X = X is plausible
X = Y is not plausible

Differentiation and integration.

dX/dY = XY-1. (a differential is of the same dimension as its base quantity)
d2X/dy2 = XY-1Y-1 = XY-2.
Integral(X dY) = XY (the integral can be seen as the area under a graph of X in terms of Y, and dY has the dimension of Y)

Logs and exponentiation

log(X) = X (the logarithm of a distance is a distance)
exp(X) = X (because exp(log(v)) = v)

Any compound such as XaYb can be substituted for X and Y above, where X and Y are different compounds in any one equation.

The addition rules above work for subtraction as well as addition.

Some examples:

An angle measured in radians can be described by the ratio of the arc of a circle to the radius of the circle, so it has dimensions L/L = 0. It is a pure number. So is an angle measured in degrees (but its scale is different when you use the equation with measured variables). So is the rate at which a clock gains or loses (seconds per day has dimension T/T = 0).

A velocity is distance per unit time, or L/T = LT-1.
An angular velocity is a number per unit time of T-1.

The equation for the radius of a circle is R = sqrt(x2 + y2) which dimensionally is (L2 + L2)-2 = (L2)-2 = L. The equation is plausible.

Non-integer exponents are allowed.
>X'*Y''-X.''*Y.dot|1/3 * R1/3 is (LT-1*LT-2 - LT-2*LT-1)1/3*L1/3 =(L2T-3* - L2T-3)1/3*L1/3 = (L2T-3)1/3*L1/3 = L/T which is a linear velocity.

dθ/ds = L-1 (angular change per unit path length, if θ is an angle and s is a distance such as a portion of a curve)
dθ/dt = T-1 (angular velocity, angular change per unit time).

d2s/dt2 (where s is a distance) yields LT-2, so force has dimension MLT-2 because F=ma.

And so forth.

Anyone can use dimensional analysis to check the plausibility of their equations. You don't have to be a mathematician. If the dimensions on opposite sides of an equal sign differ, the equation is wrong. It could be wrong even if the dimensions are the same, but the plausibility check is often a good place to start. You can also often use dimensional analysis to check whether you are starting your problem solution with a plausible set of variables. It's a quick and easy way to find some easily made errors.

Martin

--
Richard S. Marken

"The childhood of the human race is far from over. We have a long way to go before most people will understand that what they do for others is just as important to their well-being as what they do for themselves." -- William T. Powers

···

--
Richard S. Marken
"The childhood of the human race is far from over. We have a long way to go before most people will understand that what they do for others is just as important to their well-being as what they do for themselves." -- William T. Powers

Abbott · July 26, 2016, 2:46am

[From Bruce Abbott (2016.07.25.2245 EDT)]

···

Rick Marken (2016.07.25.1215) –

RM: If this is supposed to be relevant to the power law could you explain how?

BA: This is just a WILD guess, but I think Martin would like you to perform a dimensional analysis of your computations and present the results. If you are uncertain how to do this, see Martin’s tutorial below.

Bruce

[Martin Taylor 2016.07.25.13.13]:

Yet we often see equations in which similarly impossible equalities are asserted. So I thought it might be useful to offer a quick guide to dimensional analysis.

We start with a set of basic dimensions. They are basic in the sense than none can be derived from operations on the others. One possible set is distance, time, and mass, which we can symbolize as L, T, and M, respectively. If these are taken as the basic dimensions, then Force cannot be basic, since F=Ma, and acceleration is based on distance and time There are other basic dimensions such as electromagnetic ones, but L, T, and M will do for my purpose in this message.

It is impossible to add a mass to a time and get anything sensible, or to add a distance to an area, an area being one distance multiplied by another. Taking X and Y to be different dimensions (basic or compound), we have the following rules.

X + X = X (adding a distance to a distance yields a distance; adding an area to an area yields an area)
X * X = X² (multiplying a distance by a distance yields an area; multiplying X by itself n times yields Xⁿ.)
X / X = 0 (dividing a distance by a distance give a dimensionless quantity, just a number)

X+Y = BAD (you can’t add a distance to a mass)
X * Y = XY (you can multiply a distance by a mass)
X / Y = XY^-1 (you can divide a distance by a time; the result is called speed or velocity)

An equation:
X = X is plausible
X = Y is not plausible

Differentiation and integration.

dX/dY = XY^-1. (a differential is of the same dimension as its base quantity)
d²X/dy² = XY^-1Y^-1 = XY^-2.
Integral(X dY) = XY (the integral can be seen as the area under a graph of X in terms of Y, and dY has the dimension of Y)

Logs and exponentiation

log(X) = X (the logarithm of a distance is a distance)
exp(X) = X (because exp(log(v)) = v)

Any compound such as X^aY^b can be substituted for X and Y above, where X and Y are different compounds in any one equation.

The addition rules above work for subtraction as well as addition.

Some examples:

An angle measured in radians can be described by the ratio of the arc of a circle to the radius of the circle, so it has dimensions L/L = 0. It is a pure number. So is an angle measured in degrees (but its scale is different when you use the equation with measured variables). So is the rate at which a clock gains or loses (seconds per day has dimension T/T = 0).

A velocity is distance per unit time, or L/T = LT^-1.
An angular velocity is a number per unit time of T^-1.

The equation for the radius of a circle is R = sqrt(x² + y²) which dimensionally is (L² + L²)^-2 = (L²)^-2 = L. The equation is plausible.

Non-integer exponents are allowed.

X’*Y’‘-X.’'*Y.dot|^1/3 * R^1/3 is (LT^-1*LT^-2 - LT^-2*LT^-1)^1/3L^1/3 =(L²T^-3 - L²T^-3)^1/3*L1^/3 = (L²T^-3)^1/3*L^1/3 = L/T which is a linear velocity.

dθ/ds = L^-1 (angular change per unit path length, if θ is an angle and s is a distance such as a portion of a curve)
dθ/dt = T^-1 (angular velocity, angular change per unit time).

d²s/dt² (where s is a distance) yields LT^-2, so force has dimension MLT^-2 because F=ma.

And so forth.

Anyone can use dimensional analysis to check the plausibility of their equations. You don’t have to be a mathematician. If the dimensions on opposite sides of an equal sign differ, the equation is wrong. It could be wrong even if the dimensions are the same, but the plausibility check is often a good place to start. You can also often use dimensional analysis to check whether you are starting your problem solution with a plausible set of variables. It’s a quick and easy way to find some easily made errors.

Martin

–

Richard S. Marken

“The childhood of the human race is far from over. We have a long way to go before most people will understand that what they do for others is just as important to their well-being as what they do for themselves.” – William T. Powers

–

Richard S. Marken

“The childhood of the human race is far from over. We have a long way to go before most people will understand that what they do for others is just as important to their well-being as what they do for themselves.” – William T. Powers

MartinT · July 26, 2016, 4:01am

[Martin Taylor 2016.07.25.23.09]

Actually, I had a wider objective than that. I thought that there

might be some on CSGnet who had not had my luck in being introduced
to dimensional analysis as a beginning undergraduate, and that a
quick description might show how useful and easy it can be. I’m no
longer concerned with Rick’s fundamental mistake, except insofar as
it can mislead some CSGnet readers, as it did Fred. Alex earlier
gave up as a lost cause trying to show Rick how to correct his
mistake, and it took a bit longer for me to come to the same point.
But I am trying to work on Alex’s problem.
In trying to see if I can find an answer for Alex, I have at least
once saved myself from making a silly (in retrospect) mistake by
doing a dimensional analysis check on an equation. I didn’t do it
consciously, but just perceived that the two sides didn’t match in
the same way I might perceive a chair as red. I suppose it’s just a
perceptual function I have developed that produces a perception I
control. I was immediately able to see that one of Rick’s early
expressions in the curvature discussion was wrong because the two
sides of the equal sign didn’t match in dimensions. Rick fixed that
problem, and I included the example of the fixed statement as one of
my examples.
What Rick hasn’t fixed, so far as I can see, is actually a
dimensional problem, but it is not of a type I discuss in the
tutorial. The problem that has plagued the curvature threads
throughout is that he uses derivatives with respect to time but
starts with statements in which the derivatives should be with
respect to distance. He mentioned that he got help from (if I
remember correctly) his mathematician son, but the Wikipedia article
on curvature makes the error very easy to make, and maybe his son
used a similar uniform speed example to make the analysis of
curvature easier to visualize.
The Wikipedia article starts by asserting that the along track speed
(ds/dt) is one unit, and then treats that unit as a dimensionless
number, rather than as having the correct dimension LT.
Everything thereafter is then dimensionally misleading, because the
derivatives are taken with respect to time rather than taking the
numerically equal derivative with respect to distance along the
curve, thus justifying the words “angular velocity” as though the
reference is to d
θ/dt, when it should be dθ/ds in the general case of arbitrary
speed. I coined a word “telocity” for my own convenience to refer to
dx/ds in cases where dx/dt would justify the word “velocity”. Rick’s
V’s are “telocities”, whereas Alex was asking about velocities.
Treating “telocities” and velocities as the same on the grounds that
the speed along track is exactly 1 dimensionless unit is such an
easy error to make, but it completely eliminates any possibility of
addressing Alex’s problem. Keeping track of the dimensions when you
have numerical equalities can save you from that kind of error. It
did for me – as I mentioned above. When it did save me, I thought
it might be helpful to inform people how simple and useful
dimensional analysis can be (or remind them if they knew it all
along).
If Rick wants to use it, fine. It’s a tool, which comes in handy
when the situation is appropriate – when one is playing with
equations that involve physical quantities. It’s usually of no value
in analysis of control loops even though they do involve physical
quantities, because all the dimensional changes around the loop have
cancelled themselves out when you return to the starting point.
To answer Rick’s: “?”, I would say that it
resolves the dispute entirely, since it shows why he confuses
properties of speeds with properties of shapes. The power law
problem is about speeds; Rick finds a long known property of shapes
and calls it speeds.
Martin

···

On 2016/07/25 10:46 PM, Bruce Abbott
wrote:

        [From
Bruce Abbott (2016.07.25.2245 EDT)]

                Rick
Marken (2016.07.25.1215) –

                      RM: If this is supposed to
be relevant to the power law could you explain
how?

                      BA: This is just a WILD
guess, but I think Martin would like you to
perform a dimensional analysis of your
computations and present the results. If you
are uncertain how to do this, see Martin’s
tutorial below.

^-1

 If this is supposed to be relevant to the

power law could you explain how*

Bruce

                          [Martin Taylor

2016.07.25.13.13]:

Yet we
often see equations in which similarly
impossible equalities are asserted. So
I thought it might be useful to offer
a quick guide to dimensional analysis.
 We start with a set of basic
dimensions. They are basic in the
sense than none can be derived from
operations on the others. One possible
set is distance, time, and mass, which
we can symbolize as L, T, and M,
respectively. If these are taken as
the basic dimensions, then Force
cannot be basic, since F=Ma, and
acceleration is based on distance and
time There are other basic dimensions
such as electromagnetic ones, but L,
T, and M will do for my purpose in
this message.
 It is impossible to add a mass to a
time and get anything sensible, or to
add a distance to an area, an area
being one distance multiplied by
another. Taking X and Y to be
different dimensions (basic or
compound), we have the following
rules.
 X + X = X (adding a distance to a
distance yields a distance; adding an
area to an area yields an area)
X * X = X² (multiplying a
distance by a distance yields an area;
multiplying X by itself n times yields
Xⁿ.)
X / X = 0 (dividing a distance by a
distance give a dimensionless
quantity, just a number)
 X+Y = BAD (you can't add a distance to
a mass)
X * Y = XY (you can multiply a
distance by a mass)
X / Y = XY^-1 (you can
divide a distance by a time; the
result is called speed or velocity)
 An equation:
 X = X is plausible
 X = Y is not plausible

 Differentiation and integration.

 dX/dY = XY-1 . (a
differential is of the same dimension
as its base quantity)
d²X/dy² = XY^-1Y^-1
= XY^-2.
Integral(X dY) = XY (the integral can
be seen as the area under a graph of X
in terms of Y, and dY has the
dimension of Y)
 Logs and exponentiation

 log(X) = X (the logarithm of a
distance is a distance)
exp(X) = X (because exp(log(v)) = v)
 Any compound such as XaYb
 can be substituted for X and Y above,
where X and Y are different compounds
in any one equation.
 The addition rules above work for
subtraction as well as addition.
 Some examples:

 An angle measured in radians can be
described by the ratio of the arc of a
circle to the radius of the circle, so
it has dimensions L/L = 0. It is a
pure number. So is an angle measured
in degrees (but its scale is different
when you use the equation with
measured variables). So is the rate at
which a clock gains or loses (seconds
per day has dimension T/T = 0).
 A velocity is distance per unit time,
or L/T = LT^-1.
An angular velocity is a number per
unit time of T^-1.
 The equation for the radius of a
circle is R = sqrt(x² + y² )
which dimensionally is (L²
+ L²)^-2 = (L²)^-2
= L. The equation is plausible.
 Non-integer exponents are allowed.
 |X'*Y''-X.''*Y.dot|1/3
 * R1/3 is (LT-1*LT-2
 - LT-2*LT-1)1/3*L1/3
 =(L2T-3* - L2T-3)1/3*L1/3
 = (L2T-3)1/3*L1/3 
= L/T which is a linear velocity.
 dθ/ds = L-1 (angular change
per unit path length, if θ is an angle
and s is a distance such as a portion
of a curve)
dθ/dt = T^-1 (angular
velocity, angular change per unit
time).
 d2s/dt2 (where s
is a distance) yields LT^-2 ,
so force has dimension MLT^-2
because F=ma.
 And so forth.

 Anyone can use dimensional analysis to
check the plausibility of their
equations. You don’t have to be a
mathematician. If the dimensions on
opposite sides of an equal sign
differ, the equation is wrong. It
could be wrong even if the dimensions
are the same, but the plausibility
check is often a good place to start.
You can also often use dimensional
analysis to check whether you are
starting your problem solution with a
plausible set of variables. It’s a
quick and easy way to find some easily
made errors.
 Martin

–

                                              Richard

S. Marken

                                                "The
childhood of the
human race is far
from over. We have a
long way to go
before most people
will understand that
what they do for
others is just as
important to their
well-being as what
they do for
themselves." –
William T. Powers

–

                                    Richard S.

Marken

                                      "The childhood of the human
race is far from over. We have
a long way to go before most
people will understand that
what they do for others is
just as important to their
well-being as what they do for
themselves." – William T.
Powers

MartinT · July 26, 2016, 4:10am

[Martin Taylor 2016.07.26.00.03]

I know of no equation that says energy is equal to mass times a numerical
constant. I do know of one that says energy is equal to mass times a
constant of dimension L²T^-2 . As a small
exercise, why don’t you check to see whether this is dimensionally
consistent, remembering that energy is also force times distance and
f=ma. Hint it’s a one-liner if you use one of the examples in the
tutorial. You don’t need any intermediate steps.

Martin

···

[From, Rick Marken (2016.07.25.1215)]

RM Fly larva brain attacks again!

RM: Of course, I means the one that says energy is
equal to mass times a constant.

                      MT: If, say, an equation
says that a mass is equal to a distance, or a
voltage is equal to a velocity, there’s
something wrong with the equation.

                  RM: What about that one that says mass is equal

to mass times a constant?

rupert · July 27, 2016, 2:23pm

[From Rupert Young (2016.07.27 15.20)]

(Martin Taylor 2016.07.26.00.03]

  I know of no equation that says energy is equal to
mass times a numerical constant.

Didn't Albert come up with one of those?

Rupert

MartinT · July 27, 2016, 4:18pm

[Martin Taylor 2016.07.27.12.06]

[From Rupert Young (2016.07.27 15.20)]

(Martin Taylor 2016.07.26.00.03]
    I know of no equation that says energy is equal to
mass times a numerical constant.
  Didn't Albert come up with one of those?



  Rupert

I presume you refer to Einstein, and the answer, as I already

pointed out to Rick in [Martin Taylor 2016.07.26.00.03], is “No”.
The constant in question is not numerical, but is a speed squared.
Since speed has the dimension LT^-1 , speed squared has the
dimension L²T^-2 so the equation E = Mc²
has dimensionality ML²T^-² on the
right hand side.

I find it reassuring that both Newton and Einstein came up with the

same dimensionality for energy from wildly different conceptual
starting points. (I gave Rick a hint about how to find the Newton
value in the cited message).

Martin