An Introduction to Mechanics by Kleppner D., Kolenkow R.

By Kleppner D., Kolenkow R.

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If R is the vector from the origin of the unprimed coordinate system to the origin of the primed coordinate system, we have r = R + r , or alternatively, r = r − R. 14 VECTORS AND KINEMATICS We use these results to show that displacement S, a true vector, is independent of coordinate system. As the sketch indicates, S = r2 − r1 = (R + r2 ) − (R + r1 ) = r2 − r1 . 1 Motion in One Dimension Before employing vectors to describe velocity and acceleration in three dimensions, it may be helpful to review one-dimensional motion: motion along a straight line.

The accuracy of a clock driven by the pendulum depends on L remaining constant, but L can change due to thermal expansion and possibly aging effects. The problem is to find how sensitive the period is to small changes in length. If the length changes by some amount l, the new period is T = 2π g/(L + l). The change in the period is ΔT = T − T 0 = 2π g − (L + l) g . L This equation is exact but not particularly informative. It gives little insight as to how ΔT depends on the change in length l. Also, if l L, which is generally the case of interest, ΔT is the small difference of two large numbers, which makes the result very sensitive to numerical errors.

The diagram shows a vector A in the x−y plane. The projections of A along the x and y coordinate axes are called the components of A, A x and Ay , respectively. The magnitude of A is A = A x 2 + Ay 2 , and the direction of A makes an angle θ = arctan (Ay /A x ) with the x axis. Since its components define a vector, we can specify a vector entirely by its components. Thus A = (A x , Ay ) or, more generally, in three dimensions, A = (A x , Ay , Az ). Prove for yourself that A = A x 2 + Ay 2 + Az 2 .

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