Question

A pendulum swings an arc with a length equal to 15 meters. Each subsequent swing is 95% of the previous swing.

a) How far with the pendulum travel on its 6th swing?

b) How far will the pendulum swing before it essentially stops? Hint: This is an infinite geometric series.

Show all work and explain your reasoning.

Answer 1

a) It will travel approx 79.47 meters,

b) It will travel 300 meters.

Explanation:

Given,

The initial distance travel on first swing = 15 meters,

Also, Each subsequent swing is 95% of the previous swing.

Thus, there is a G.P. that shows this situation,

Having first term, a = 15,

And, the common difference, r = 95 % = 0.95,

a) Also, for the 6th swing,

Number of terms, n = 6,

Hence, the distance covered by the pendulum on its 6th swing,

`S_(n)=(a(1-r^n))/(1-r)`

`S_(6)=(15(1-0.95^6))/(1-0.95)`

`=79.4724328125\approx 79.47\text{ meters}`

b) When
`n=\infty`

The distance will the pendulum swing before it essentially stops is,

`S_(\infty)=(a)/(1-r)`

`=(15)/(1-0.95)`

`=300\text{ meters}`

Answer 2

The answers for your two questions are the following

a)11.606 m

b) As the number of swings approaches infinity

Explanation:

We are dealing with an exponential equation series, where the length of each swing can be represented as

l = [ 15 *(0.95)^(n-1) ]

n is the corresponding number of the swing.

So, for the first swing, n = 1

[ 15 *(0.95)^(1-1) ] = 15 m

a) How far with the pendulum travel on its 6th swing?

We just need to evaluate the previous formula for n = 6

[ 15 *(0.95)^(6-1) ] =

[ 15 *(0.95)^(5) ] =

[ 15 *(0.7737) ] =

[ 15 *(0.7737) ] = 11.606 m

b) How far will the pendulum swing before it essentially stops? Hint: This is an infinite geometric series.

We previously stated that the length of the arc of each swing can be represented as

l(n) = [ 15 *(0.95)^(n-1) ] , for n>=1

Since the function approaches zero if and only if n approaches infinity, we can say that the pendulum never stops.

Of course, this only happens mathematically, we can always fin a threshold for which the movement cannot be registered anymore.

Please see attached graph for a representation of the function