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Chapter 8: Problem 62

Show that the form \(\Delta U=m g \Delta r\) follows from Equation 8.5 when \(r_{1} \simeq r_{2} .\) [Hint: Write \(r_{2}=r_{1}+\Delta r\) and apply the binomial approximation (Appendix A).]

Short Answer

Expert verified
After applying binomial theorem to equation 8.5 by substituting \(r_2 = r_1 + \Delta r\), we arrive at the given expression \(\Delta U = m*g*\Delta r\)

Step by step solution

01

Substitute for r2

We first express \(r_2\) in terms of \(r_1\) and \(\Delta r\) by substituting \(r_2 = r_1 + \Delta r\) into Equation 8.5.
02

Identify Binomial Term

As Equation 8.5 should contain term in form of \(1+n\) and here n should be \(\Delta r/ r_1\), which is small and hence suitable for binomial approximation.
03

Apply Binomial Approximation

Applying the binomial approximation to simplify, we replace the term \((1 + \Delta r/ r_1)^n\) with \(1 + n*\Delta r/ r_1\).
04

Simplify the Result

On simplifying the result, we obtain the expression for \(\Delta U\) in terms of \(m\), \(g\) and \(\Delta r\) as \(\Delta U = m*g*\Delta r\)

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