Using Taylor Series to Approximate Force in Gravitational Fields

In summary, the conversation discusses the relationship between force (F), gravity (g), and the radius of the Earth (R), in the situation where the distance (h) from the surface of the Earth is much smaller than the radius. It is shown that F is approximately equal to mg, and it can also be expressed as mg multiplied by a series in h/R. The use of Taylor series is suggested to find a more accurate approximation. The conversation also poses a question about the first order correction and its effect on the estimate of F.
  • #1
apiwowar
96
0
so F = mgR2/(R+h)2

where R is the radius of the earth. consider the situation where h is much smaller than R.

a) show that F is approximately equal to mg

b)express F as mg multiplied by a series in h/R

so i need help on getting started.

would showing that F is approximately equal to mg be the same as expressing F as mg multiplied by a series in h/R since taylor series are used as approximations?

i can pull an R2 out of the denominator and get rid of the R2 on the top and be left with mg(1+h/R2)-2 and in that case i could use the binomial expansion of a taylor series. would that solve a or b?

any other help would be appreciated
 
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  • #2
Yes, expanding F as a taylor series around h=0 is exactly what they are asking for.
 
  • #3
therefore a and b would be the same answer? just an expansion of the taylor series around h=0?
 
  • #4
apiwowar said:
therefore a and b would be the same answer? just an expansion of the taylor series around h=0?

You consider h to be small, so, yes, when h is near zero F is near mg. You don't need taylor series for the first one, if h is small compared with R you can just just put h=0 into the expression to find approximately what F is when h is small. You would use the taylor series to find out how good the approximation is. I'm not really sure what your question is. What they want in b is pretty clear, isn't it?
 
  • #5
there is a part c to this question asking how far above the surface of the Earth you can go before the first order correction changes the estimate f ~ mg by more than 10% (and assume that R=6400km)

so for that i just plug in .10 for h and 6400km for R into the first order approximation, i did so and got .9996875mg. is that saying that you can go 99.99% away from the surface before the first order estimate is changed by more than 10%?
 
  • #6
No. You want to calculate h from the series. If F changes by 10%, then mg changes to 0.9*mg. Figure out the value of the first order correction that will change mg (at h=0) to 0.9*mg.
 
  • #7
BTW mg*(1+h/R^2)^(-2) should be mg*(1+h/R)^(-2). Check the algebra.
 

FAQ: Using Taylor Series to Approximate Force in Gravitational Fields

What is the Taylor series used for?

The Taylor series is a mathematical tool used to represent a function as an infinite sum of its derivatives. It is often used to approximate functions and calculate values at points where the function is not defined or difficult to evaluate.

How is the Taylor series derived?

The Taylor series is derived using the Taylor polynomial, which is a finite sum of terms that approximate the function at a given point. By taking the limit as the number of terms approaches infinity, the Taylor series is obtained.

What is the difference between a Taylor series and a Maclaurin series?

A Taylor series is centered at any point on the function, while a Maclaurin series is centered at 0. This means that the coefficients in a Maclaurin series are calculated using the derivatives at 0, while in a Taylor series, they are calculated using the derivatives at the center point.

How accurate is the Taylor series approximation?

The accuracy of the Taylor series approximation depends on the function and the number of terms used. In general, the more terms included in the series, the more accurate the approximation will be. However, for some functions, the series may converge to the exact value at an infinite number of terms.

How is the Taylor series used in real-world applications?

The Taylor series has many real-world applications in fields such as physics, engineering, and economics. It can be used to solve differential equations, approximate functions and values, and analyze the behavior of various systems. It is also widely used in computer science and digital signal processing.

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