Newton's Law of Gravitation 2. The magnitude of the acceleration of an object under the pull of Earth's gravity is given by Newton's Universal Law of Gravitation a = GME R? where G is the universal gravitational constant, MẸ is the mass of Earth, and R is the distance of the object from the center of Earth. Let x be the distance above Earth's surface. We can rewrite the formula for the acceleration as a function of x by noting that R= RE + x, where RE is the radius of Earth. Therefore, a(x) = G (Re + x)² d (a) Show that dx 1 1 %3D (1 – x)²° - T. (b) Use the above fact, along with the power series of 1 to determine a power 1- x 1 series for (1+x)²° (c) What is the radius of convergence for the series in part (b)? (Hint: You do not need to calculate anything. What is the radius of convergence for the power series 1 of 1- x does not change the radius of convergence.) -? This series has the same radius of convergence since taking a derivative
Newton's Law of Gravitation 2. The magnitude of the acceleration of an object under the pull of Earth's gravity is given by Newton's Universal Law of Gravitation a = GME R? where G is the universal gravitational constant, MẸ is the mass of Earth, and R is the distance of the object from the center of Earth. Let x be the distance above Earth's surface. We can rewrite the formula for the acceleration as a function of x by noting that R= RE + x, where RE is the radius of Earth. Therefore, a(x) = G (Re + x)² d (a) Show that dx 1 1 %3D (1 – x)²° - T. (b) Use the above fact, along with the power series of 1 to determine a power 1- x 1 series for (1+x)²° (c) What is the radius of convergence for the series in part (b)? (Hint: You do not need to calculate anything. What is the radius of convergence for the power series 1 of 1- x does not change the radius of convergence.) -? This series has the same radius of convergence since taking a derivative
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Transcribed Image Text:Newton's Law of Gravitation
2. The magnitude of the acceleration of an object under the pull of Earth's gravity is
given by Newton's Universal Law of Gravitation
МЕ
a = G
R?
where G is the universal gravitational constant, ME is the mass of Earth, and R is the
distance of the object from the center of Earth.
Let x be the distance above Earth's surface. We can rewrite the formula for the
acceleration as a function of x by noting that R = Rp + x, where Rp is the radius of
Earth. Therefore,
МЕ
a(x) = G-
(RE + x)2
d.
(a) Show that
dx
1
1
(1 – x)*
- x.
(b) Use the above fact, along with the power series of
1
to determine a power
1- x
1
series for
(1+x)²*
(c) What is the radius of convergence for the series in part (b)? (Hint: You do not
need to calculate anything. What is the radius of convergence for the power series
of
1
does not change the radius of convergence.)
-? This series has the same radius of convergence since taking a derivative
Transcribed Image Text:(d) Rewrite a(x) as
GME
1
a(x)
R (1+x/RE)?
and use the information from part (b) to determine a power series for a(x).
(e) What is the radius of convergence, in terms of x, for the power series you found
in part (d)? (Hint: Use part (c).)
(f) The constants have the values G = 6.674 × 10-11 m³kg¬'s-2, ME = 5.972 × 1024
kg, and RE = 6.371 × 106 m. Substitute these values into the first term of the
series you found in part. This is the zeroth degree Taylor polynomial, a T,(x).
The resulting value might be familiar. What is this value?
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