A pendulum bob swings 5.0cm on its first oscillation.On each subsequent oscillation the bob travels 2/5 of the previous distance. Find the total distance the bob travels before coming to rest.
Ah, the never-ending dance of the pendulum! Let me calculate that for you.
On the first oscillation, the bob swings 5.0cm.
On the second oscillation, it travels 2/5 of the previous distance. So, 2/5 x 5.0cm = 2.0cm.
On the third oscillation, it travels 2/5 of the previous distance again. So, 2/5 x 2.0cm = 0.8cm.
This pattern continues, getting smaller and smaller with each oscillation, until at a certain point the bob finally comes to rest.
To find the total distance the bob travels before resting, you can add up all the distances:
5.0cm + 2.0cm + 0.8cm + ... = ???
Hmm, it seems that the bob will keep going and going, getting closer and closer to zero but never truly reaching it. So, I'm sorry to say, but the total distance the bob travels before coming to rest is infinite. Just like my collection of clown noses! 🤡
To find the total distance the pendulum bob travels before coming to rest, we need to determine the sum of an infinite geometric series.
The first term, a, is 5.0 cm, and the common ratio, r, is 2/5 (since the bob travels 2/5 of the previous distance on each subsequent oscillation).
The formula to find the sum of an infinite geometric series is:
S = a / (1 - r)
Substituting the given values:
S = 5.0 cm / (1 - 2/5)
Simplifying:
S = 5.0 cm / (3/5)
S = 25/3 cm
Therefore, the total distance the pendulum bob travels before coming to rest is 25/3 cm.
To find the total distance the bob travels before coming to rest, we can determine the sum of an infinite geometric series. The formula for the sum of an infinite geometric series is:
S = a / (1 - r)
Where 'S' is the sum, 'a' is the first term, and 'r' is the common ratio.
In this case, the first term 'a' is 5.0 cm and the common ratio 'r' is 2/5.
Plugging the values into the formula, we have:
S = 5.0 / (1 - 2/5)
To simplify the fraction, we find the common denominator:
S = 5.0 / (5/5 - 2/5)
S = 5.0 / (3/5)
Dividing the numerator by the denominator:
S = 5.0 * (5/3)
S = 25/3 cm
The total distance the bob travels before coming to rest is 25/3 cm.
total swing distance
= 5 + (2/5)(5) + (2/5)^2 (5 ) + ...
this is a GS where a = 5 and r = 2/5
sum(infinite terms) = a/(1-r)
= 5/(1-2/5)
= 5/(3/5)
= 5(5/3)
= 25/3