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workenergy1

# workenergy1 - Math 21a Supplement 1 on Work and Energy...

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Math 21a Supplement 1 on Work and Energy Newton’s law asserts that the position vector r (t) of a particle of mass m under the influence of a force F obeys the equation m r ´´ = F . (1) a) Work Suppose that the components of the force vector F do not depend on time or the position of the 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 r 0 to position r 1 is (by definition) W F •( r 1 - r 0 ). (2) Note that this notion of work can be generalized to apply to any force vector F , constant or not; but we are not ready at this point in the course for the generalization. b) Energy The energy of a particle at position r with velocity vector r ´ is the function e 2 1 m | r ´| 2 - F r . (3) Note that Newton’s law (Equation (1)) implies that the energy function does not change as time evolves. Indeed, we can differentiate (3) to find that e ´ = m r ´´• r ´ - F r ´ = F r ´ - F r ´ = 0 , (4)

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