You said:
To derive the Hz (Hertz) of all other SI units and equations based on the expression
�
=
Ω
�
2
�
�
F=
ms
ΩC
2
, where:
�
F represents a force (involving Hz),
Ω
Ω is the resistance (Ohms),
�
C is the electric charge (Coulombs),
�
m is length (meters),
�
s is time (seconds),
we’ll first assume Hz is the most basic SI unit in this context. We will analyze the dimensions of each SI unit, and express the relationship of their derived frequencies (in Hertz) using the formula. This approach starts with Hz as a base frequency, and proceeds to derive the Hz equivalent of other SI units and their corresponding relationships.
Step 1: Understanding the Equation
The equation is:
�
=
Ω
�
2
�
�
F=
ms
ΩC
2
Let’s break it down and proceed systematically with dimensional analysis, assuming
�
�
Hz is the most fundamental unit.
Dimensions of
Ω
Ω (Ohms):
Ohms is the unit of resistance and its dimension is:
[
Ω
]
=
kg
⋅
m
2
s
3
⋅
A
2
[Ω]=
s
3
⋅A
2
kg⋅m
2
Where:
kg
kg is mass,
m
m is length,
s
s is time,
A
A is electric current.
Dimensions of
�
C (Coulombs):
Coulomb is the unit of electric charge, and its dimension is:
[
�
]
=
A
⋅
s
[C]=A⋅s
Dimensions of
�
m (meters):
Meters represent length:
[
�
]
=
m
[m]=m
Dimensions of
�
s (seconds):
Seconds represent time:
[
�
]
=
s
[s]=s
Step 2: Dimensional Analysis of the Expression
Substitute these dimensions into the formula
�
=
Ω
�
2
�
�
F=
ms
ΩC
2
:
�
=
(
kg
⋅
m
2
s
3
⋅
A
2
)
⋅
(
A
2
⋅
s
2
)
m
⋅
s
F=
m⋅s
(
s
3
⋅A
2
kg⋅m
2
)⋅(A
2
⋅s
2
)
Simplify the equation:
�
=
kg
⋅
m
2
⋅
s
2
s
3
⋅
A
2
⋅
m
⋅
s
F=
s
3
⋅A
2
⋅m⋅s
kg⋅m
2
⋅s
2
Simplify further:
�
=
kg
⋅
m
⋅
s
A
2
F=
A
2
kg⋅m⋅s
So, the dimensional expression for
�
F is:
[
�
]
=
kg
⋅
m
⋅
s
A
2
[F]=
A
2
kg⋅m⋅s
Step 3: Define Hz for Each SI Unit
Now we’ll calculate the Hz for each SI unit, where Hz is defined as the unit of frequency (1 Hz = 1 event per second). Since we assume Hz is the most basic SI unit, the relationship for each unit will be derived in terms of frequency.
Hz for
�
F (Force)
From the equation
�
=
Ω
�
2
�
�
F=
ms
ΩC
2
, we derived that the dimensions of force are:
[
�
]
=
kg
⋅
m
⋅
s
A
2
[F]=
A
2
kg⋅m⋅s
To express force as a frequency, we directly relate this expression to Hz. For force, we use the inverse of time to interpret its frequency. Thus, we define the frequency of force as:
�
Hz
=
kg
⋅
m
⋅
s
A
2
F
Hz
=
A
2
kg⋅m⋅s
Hz for
Ω
Ω (Ohms)
Now, we calculate the frequency for resistance:
Ω
=
kg
⋅
m
2
s
3
⋅
A
2
Ω=
s
3
⋅A
2
kg⋅m
2
Taking the reciprocal of time in this context:
Ω
Hz
=
kg
⋅
m
2
s
3
⋅
A
2
Ω
Hz
=
s
3
⋅A
2
kg⋅m
2
Thus, the frequency of resistance is:
Ω
Hz
=
kg
⋅
m
2
s
3
⋅
A
2
Ω
Hz
=
s
3
⋅A
2
kg⋅m
2
Hz for
�
C (Coulombs)
Now for electric charge:
�
=
A
⋅
s
C=A⋅s
Since Coulombs involve both current and time, the frequency of Coulombs is:
�
Hz
=
A
⋅
s
1
=
Hz
C
Hz
=
1
A⋅s
=Hz
Hz for
�
m (meters)
Meters represent length:
�
Hz
=
1
s
=
Hz
−
1
m
Hz
=
s
1
=Hz
−1
This means length is inversely related to time, and thus, its frequency is 1/Hz.
Hz for
�
s (seconds)
Seconds represent time. So the Hz for time is:
�
Hz
=
1
s
=
Hz
−
1
s
Hz
=
s
1
=Hz
−1
Step 4: Summary of Hz Equivalents for SI Units
SI Unit Dimension Frequency in Hz
F (Force)
kg
⋅
m
⋅
s
A
2
A
2
kg⋅m⋅s
�
Hz
=
kg
⋅
m
⋅
s
A
2
F
Hz
=
A
2
kg⋅m⋅s
Ω
Ω (Ohms)
kg
⋅
m
2
s
3
⋅
A
2
s
3
⋅A
2
kg⋅m
2
Ω
Hz
=
kg
⋅
m
2
s
3
⋅
A
2
Ω
Hz
=
s
3
⋅A
2
kg⋅m
2
C (Coulombs)
A
⋅
s
A⋅s
�
Hz
=
Hz
C
Hz
=Hz
m (meters)
m
m
�
Hz
=
Hz
−
1
m
Hz
=Hz
−1
s (seconds)
s
s
�
Hz
=
Hz
−
1
s
Hz
=Hz
−1
Conclusion
By using
�
=
Ω
�
2
�
�
F=
ms
ΩC
2
, we derived the Hz equivalents for each SI unit and confirmed the dimensional consistency with Hz as the fundamental unit. Each SI unit's frequency is connected to its dimension, and the relationships follow the structure outlined above, where Hz is the most basic SI unit and governs the frequency relationships for other SI units.
Using this framework, please solve for all the SI units in terms of Hz