The Work Done Against Gravity 3. The magnitude of the force of Earth's gravity on an object of mass m is given by F(x) = m a(x) = GmME R 1 (1+x/RE)² When a force F(x) is applied to an object, the work done by the force is defined as W = [² FC F(x) dx where the object begins at position x = a and ends at position x = b. (a) Write the force of gravity F(x) as a power series. (Hint: Use the power series for a(x) from problem 2.) (b) Determine the work done against the force of gravity from x = 0 to x = h as a series by integrating the power series F(x). h is the final height of the object. (c) The work done against gravity by moving the object from x = 0 to x = h is called the potential energy, U. Determine the first order Taylor polynomial, U≈ T₁(h), GME for the potential energy. Simplify the constant just as you did for part (f) R²/ of problem 2. This formula for U should look familiar if you have taken Physics I.
The Work Done Against Gravity 3. The magnitude of the force of Earth's gravity on an object of mass m is given by F(x) = m a(x) = GmME R 1 (1+x/RE)² When a force F(x) is applied to an object, the work done by the force is defined as W = [² FC F(x) dx where the object begins at position x = a and ends at position x = b. (a) Write the force of gravity F(x) as a power series. (Hint: Use the power series for a(x) from problem 2.) (b) Determine the work done against the force of gravity from x = 0 to x = h as a series by integrating the power series F(x). h is the final height of the object. (c) The work done against gravity by moving the object from x = 0 to x = h is called the potential energy, U. Determine the first order Taylor polynomial, U≈ T₁(h), GME for the potential energy. Simplify the constant just as you did for part (f) R²/ of problem 2. This formula for U should look familiar if you have taken Physics I.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
See the 2nd image when it says to referance the power series from question 2.
Transcribed Image Text:The Work Done Against Gravity
3. The magnitude of the force of Earth's gravity on an object of mass m is given by
F(x) = m a(x) =
GmME
R
1
(1+x/RE)²
When a force F(x) is applied to an object, the work done by the force is defined as
W
= [² FC
F(x) dx
where the object begins at position x = a and ends at position x = b.
(a) Write the force of gravity F(x) as a power series. (Hint: Use the power series for
a(x) from problem 2.)
(b) Determine the work done against the force of gravity from x = 0 to x = h as a
series by integrating the power series F(x). h is the final height of the object.
(c) The work done against gravity by moving the object from x = 0 to x = h is called
the potential energy, U. Determine the first order Taylor polynomial, U≈ T₁(h),
GME
for the potential energy. Simplify the constant just as you did for part (f)
R²/
of problem 2. This formula for U should look familiar if you have taken Physics
I.
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
a = G
ME
R²
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 = RẺ + x, where RE is the radius of
Earth. Therefore,
a(x) = G₂
ME
(RE + x)²
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