Question

A pendulum swing through an arc of length 54cm. On each successive swing, the length
of the arc is 0.92 of the previous length.
(a) What is the length of the arc after 7 swings.
(b) At which swing is the length of the arc of the pendulum less than 17cm for the
first time.
(c) Find the total distance covered by the pendulum after 22 swings.
(d) Find the total distance covered by the pendulum before it comes to a stop.

Answer 1

Let $l_n$ be the length of the arc after $n$ swings. Then $l_n=54\cdot (0.92)^n.$

(a) After 7 swings the length of the arc is $l_7=54\cdot(0.92)^7\approx 30.12$ (cm)

(b) Since $l_{13}=54\cdot(0.92)^{13}\approx 18.27$ and $l_{14}=54\cdot(0.92)^{14}\approx 16.8$ (cm), we conclude that at 14 swing the length of the arc of the pendulum is less than 17cm for the first time.

(c) Find the total distance covered by the pendulum after 22 swings as the sum of a geometric progression with common ratio $0.92$: $S_{22}=54\cdot \frac{1-0.92^{22}}{1-0.92}\approx 567.2$ (cm).

(d) Find the total distance covered by the pendulum before it comes to a stop as the sum of an infinite geometric progression with common ratio $0.92<1$: $S=54\cdot \frac{1}{1-0.92}=675$ (cm).