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HARVARD MATH 21A - Supplement 1 on Work and Energy

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Math 21a Supplement 1 on Work and EnergyNewton’s law asserts that the position vector r(t) of a particle of mass m under theinfluence of a force F obeys the equationm r´´ = F .(1)a) WorkSuppose that the components of the force vector F do not depend on time or the position ofthe particle. Thus, F has components (a, b, c) which are numbers, not functions. (For example,F = (1, 2, 3).) Then, the work done in moving the particle from position r0 to position r1 is (bydefinition)W ≡ F•(r1 - r0).(2)Note that this notion of work can be generalized to apply to any force vector F, constant or not; butwe are not ready at this point in the course for the generalization.b) EnergyThe energy of a particle at position r with velocity vector r´ is the functione ≡ 2−1 m |r´|2 - F•r .(3)Note that Newton’s law (Equation (1)) implies that the energy function does not change as timeevolves. Indeed, we can differentiate (3) to find thate´ = m r´´•r´ - F•r´ = F•r´ - F•r´ = 0 ,(4)where the second equality comes by substituting for m r´´ using Equation (1).Note that the constant force vector is not the only kind of force for which energy can bedefined and for which the energy is independent of time along the trajectory. Consider, forexample the following case: Let f(r) be a function of the distance, r ≡ |r|, of the particle from theorigin, and consider the force F = f(r) r/r. For example, if a force is gravitational and due to amass M at the origin, then f(r) = - G m M r−2, where G is the Gravitational constant . In any event,when F = f(r) r/r, Newton’s law readsm r´´ = f(r) r/r .(5)In this case, the energy is defined to bee ≡ 2−1 m |r´|2 - V(r) ,(6)where V(r) is an anti-derivative of f(r). That is, dVdr = f. For the case where f = G m M r−2, thisfunction V(r) (called the potential ) can be taken to be V(r) = G M m r−1.In any case, with F = f(r) r/r, the energy is also constant along a particle’s trajectory as canbe seen by first differentiating and using the Chain rule to find thate´ = m r´´•r´ - f(r) r´ .(7)One then employs the formula r´ = r´•r/r (which is also an application of the Chain rule togetherwith the fact that r = (r•r)1/2). This allows (7) to be written ase´ = m r´´• r´- f(r) r´•r/r .(8)Finally, use Equation (5) to replace m r´´ in this last equation by f(r) r/r to see that e´


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HARVARD MATH 21A - Supplement 1 on Work and Energy

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