   Chapter 11, Problem 60RE

Chapter
Section
Textbook Problem

# The force due to gravity on an object with mass m at a height h above the surface of the earth is F = m g R 2 ( R + h ) 2 where R is the radius of the earth and g is the acceleration due to gravity for an object on the surface of the earth. (a) Express F as a series in powers of h/R. (b) Observe that if we approximate F by the first term in the series, we get the expression F ≈ mg that is usually used when h is much smaller than R. Use the Alternating Series Estimation Theorem to estimate the range of values of h for which the approximation F ≈ mg is accurate to within one percent. (Use R = 6400 km.)

(a)

To determine

To express: The surface of the earth F as a series in power of hR.

Explanation

Given:

The surface of the earth is F=mgR2(R+h)2.

Result used:

If k is any real number and |x|<1, then the binomial series is

(1+x)k=n=0(kn)xn=1+kx+k(k1)2!x2+k(k1)(k2)3!x3+ (1)

Calculation:

Consider the binomial series in F=mgR2(R+h)2 as shown below.

F=mgR2(R+h)2=mgR2(R(1+hR))2=mgR2R2(1+hR)2=mg(1+hR)2

That is, F=mg(1+hR)2.

Substitute hR for x and −2 for k in equation (1),

(1+hR)2=n=0(2n)(hR)n

The surface of earth F becomes,

F=m

(b)

To determine

To estimate: The range of values of h for which the approximation Fmg is accurate to within 1 percent.

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