Considering the Principle of Homogeneity of Dimensions, is the following equation correct? F=mh, where F = force , and h = height.

Sunday, June 9, 2013

Considering the Principle of Homogeneity of Dimensions, is the following equation correct? F=mh, where F = force , and h = height.

The Principle of Homogeneity of Dimensions states that all terms inside a physical formula must be the same. Really, it is a bad idea to add meters to kilograms (multiplying meters by kilograms is okay).

There are only two terms in our formula $\displaystyle{F}={m}\cdot{h},$ $\displaystyle{F}$ and $\displaystyle{m}\cdot{h}.$ We have to check and compare their dimensions.

Force $\displaystyle{F}$ has the dimension of Newtons ( $\displaystyle{N}$ ). $\displaystyle{N}$ is the same as $\displaystyle{k}{g{\cdot}}\frac{{m}}{{{s}}^{{2}}}$ ; we can derive this from Newton's Second Law $\displaystyle{F}={m}{a}.$ Height $\displaystyle{h}$ has the dimension of meters ( $\displaystyle{m}$ ), and mass $\displaystyle{m}$ has the dimension of $\displaystyle{k}{g{.}}$ In terms of dimensional formulas $\displaystyle{F}={\left[{M}\cdot{L}\cdot{{T}}^{{-{2}}}\right]},$ $\displaystyle{m}={\left[{M}\right]}$ and $\displaystyle{h}={\left[{L}\right]}.$

So there is $\displaystyle{\left[{M}\cdot{L}\cdot{{T}}^{{-{2}}}\right]}$ at the left and $\displaystyle{\left[{M}\cdot{L}\right]}$ at the right. These dimensions are different, so no, this formula cannot be correct.

Ninenotes

Sunday, June 9, 2013

Considering the Principle of Homogeneity of Dimensions, is the following equation correct? F=mh, where F = force , and h = height.

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