Problems are listed in approximate order of difficulty. A single dot (•) indicates straightforward problems involving just one main concept and sometimes requiring no more than substitution of numbers in the appropriate formula. Two dots (••) identify problems that are slightly more challenging and usually involve more than one concept. Three dots (•••) indicate problems that are distinctly more challenging, either because they are intrinsically difficult or involve lengthy calculations. Needless to say, these distinctions are hard to draw and are only approximate.
•• In deriving the work-energy theorem (Example 2.9 in Section 2.7), we used the chain rule to find the derivative dE/dt in Eq. (2.47). This is actually a rather subtle point because E is a function of three variables, E(px, py, pz), and the relevant form of the chain rule is
where the three derivatives are partial derivatives. Use this equation to fill in the details in Eq. (2.47). (If you don’t know about partial derivatives, don’t try this problem now. We will describe partial differentiation carefully in Chapter 8.)
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