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guapka [62]
3 years ago
6

Based on the following equation, answer the questions below. ρ = (2γϕ + ψ)/rg where ρ [=] moles per cubic foot [mol/ft3] γ [=] j

oules per kilogram [J/kg] r [=] inches [in] g [=] meters per second squared [m/s^2] The number 2 is unitless and dimensionless. What are the fundamental dimensions of ψ? What are the fundamental dimensions of ϕ?
Physics
1 answer:
AlekseyPX3 years ago
7 0

1) Fundamental units of \Psi are [\frac{mol}{m\cdot s^2}]

2) Fundamental units of \Phi are [\frac{mol}{m^3}]

Explanation:

The equation for the variable \rho is

\rho =\frac{2\gamma \Phi+\Psi}{rg}

where we have:

\rho measured in [\frac{mol}{ft^3}]

\gamma measured in [\frac{J}{kg}]

r measured in [in]

g measured in [\frac{m}{s^2}]

We can re-write the equation as

\rho rg = 2\gamma \Phi + \Psi

And we notice that the units of the term on the left must be equal to the units of the term on the right.

This means that:

1) First of all, \Psi must have the same units of \rho r g. So,

[\rho r g]=[\frac{mol}{ft^3}][in][\frac{m}{s^2}]

However, both ft (feet) and in (inches) are not fundamental dimensions: this means that they can be expressed as meters. Therefore, the fundamental units of \Psi are

[\Psi]=[\frac{mol}{m^3}][m][\frac{m}{s^2}]=[\frac{mol}{m\cdot s^2}]

2)

The term 2\gamma \Phi must have the same units of \Psi in order to be added to it. Therefore,

[\gamma \Phi] = [\frac{mol}{m\cdot s^2}]

We also know that the units of \gamma are [\frac{J}{kg}], therefore

[\frac{J}{kg}][\Phi]= [\frac{mol}{m\cdot s^2}]

And so, the fundamental units of \Phi are

[\Phi]= [\frac{mol\cdot kg}{J\cdot m\cdot s^2}]

However, the Joules can be written as

[J]=[kg][\frac{m^2}{s^2}]

Therefore

[\Phi]= [\frac{mol\cdot kg}{(kg \frac{m^2}{s^2})\cdot m\cdot s^2}]=[\Phi]= [\frac{mol}{m^3}]

#LearnwithBrainly

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