Calculate gravitational acceleration in a vacuum?

[rockchalk1312]

The gravitational acceleration in vacuum varies with latitude, elevation, and local inhomogeneities in the Earth's crust. Use the following equation to find g in Lawrence based on a latitude of λ=38°57' ±3' and an elevation of H=259±10m. Calculate the uncertainty using the total difference.g = 9.780556m/s²[1+0.0052885sin²λ-0.0000059sin²(2λ)]-0.0000020s^-2HHow do you include latitude when it's at three different dimensions? And I know the total difference formula but how would you put it in here exactly? Thank you very much for any help!

 

[cepheid]

The gravitational acceleration in vacuum varies with latitude, elevation, and local inhomogeneities in the Earth's crust. Use the following equation to find g in Lawrence based on a latitude of λ=38°57' ±3' and an elevation of H=259±10m. Calculate the uncertainty using the total difference.


g = 9.780556m/s²[1+0.0052885sin²λ-0.0000059sin²(2λ)]-0.0000020s^-2H


How do you include latitude when it's at three different dimensions? And I know the total difference formula but how would you put it in here exactly? Thank you very much for any help!

The latitude is being expressed in degrees (°), arcminutes ('), and arcseconds ("), where:

1 arcmin = (1/60) degrees

1 arcsec = (1/60) arcmins

 

[haruspex]

I assume that's a formula you've been given, so it's obvious how to use the latitude and altitude in computing g. That leaves me not knowing what you're asking. Are you puzzled as to how the formula can be valid?
When asked to compute an uncertainty in f=f(x,y,..), given the uncertainties in x, y..., there are two principal methods producing different answers: absolute uncertainty and statistical uncertainty. The first is appropriate when the given uncertainties are absolute uncertainties with uniform distributions (as often is the case with instrumental measurements), there are only two or three parameters, and the answer is to be relied on absolutely, as may be the case in structural engineering. In this method, just plug in all the combinations of extreme values for the parameters and see what f values come out.
For statitical uncertainty you assume the given uncertainties are some number of standard deviations, the same number in each parameter, or otherwise convert to that form. You may then be able to use a root-sum-square formula to combine them, and the answer will represent the same number of standard deviations for f.
E.g. if f=x*y, Δf = xΔy+yΔx; E[(Δf)²] = E[x²]Δy²+E[y²]Δx²; σ(f)/f = √((σ(x)/x)²+(σ(y)/y)²).
More generally, Δf = (∂f/∂x)Δx+(∂f/∂y)Δy, etc.