By Kleppner D., Kolenkow R.

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You might feel that there is something essentially wrong with substituting inexact results for exact ones but this is not really the case, as the following example illustrates. The period of a simple pendulum of length L is T 0 = 2π g/L, where g is the acceleration of gravity. ) 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.

Find a vector A from the origin to a point on the line between r1 and r2 at distance xr from the point at r1 where x is some number. 13 Expressing one vector in terms of another Let A be an arbitrary vector and let nˆ be a unit vector in some fixed ˆ nˆ + (nˆ × A) × n. ˆ direction. 14 Two points Consider two points located at r1 and r2 , and separated by distance r = |r1 −r2 |. Find a time-dependent vector A(t) from the origin that is at r1 at time t1 and at r2 at time t2 = t1 + T . Assume that A(t) moves uniformly along the straight line between the two points.

In the limit Δt → 0, Δr becomes tangent to the trajectory, as the sketch indicates. The relation dr Δr ≈ Δt dt = vΔt becomes exact in the limit Δt → 0, and shows that v is parallel to Δr; the instantaneous velocity v of a particle is everywhere tangent to the trajectory. 7 Finding Velocity from Position Suppose that the position of a particle is given by r = A(eαt ˆi + e−αt ˆj), where A and α are constants. Find the velocity, and sketch the trajectory. dr dt = A(αeαt ˆi − αe−αt ˆj) v= or v x = Aαeαt vy = −Aαe−αt .