Because of friction and air resistance, each swing of a pendulum is a little shorter than the previous one. The lengths of the swings form a geometric sequence. Suppose the first swing of the pendulum has an arc length of 100 cm and a return swing of 99 cm.
a.)On which swing will the length first have a length less than 50 cm?
b.)Find the total distance traveled by the pendulum until it comes to rest.
To find the answer to this problem, we need to calculate the lengths of each swing in the geometric sequence and then determine when the length first becomes less than 50 cm. Additionally, we can find the total distance traveled by summing up the lengths of all the swings until the pendulum comes to rest.
Here's how we can approach both parts:
a.) Finding the swing with a length less than 50 cm:
1. We know that each swing is a little shorter than the previous one, which means that we have a decreasing geometric sequence.
2. Let's denote the length of the first swing as a₁ = 100 cm.
3. Since the return swing is one unit shorter, we can consider it as a first term of the sequence, so a₂ = 99 cm.
4. To find the common ratio (r) between the terms of the sequence, we can divide any term by its previous term. In this case, we can use a₂/a₁ = 99/100.
This gives us r = 99/100.
Now, let's find which term has a length less than 50 cm:
We have the formula for the nth term of a geometric sequence: aₙ = a₁ * r^(n-1).
We need to solve for n, where aₙ < 50.
a₁ * r^(n-1) < 50.
Substituting the values we know, we have:
100 * (99/100)^(n-1) < 50.
Simplifying the equation, we get:
(99/100)^(n-1) < 1/2.
To solve for n, we can take the logarithm (base 99/100) of both sides:
(n-1) * log(99/100) < log(1/2).
Now, we can isolate n by dividing both sides by log(99/100):
n-1 < log(1/2) / log(99/100).
Using a calculator to evaluate the right-hand side gives us:
n-1 < -0.0202.
Adding 1 to both sides gives us:
n < 0.9798.
Since n represents the term number, it must be a positive whole number, so the first term less than 1 is 1.
Therefore, the length of the pendulum swing will first have a length less than 50 cm on the 2nd swing.
b.) Finding the total distance traveled by the pendulum:
To find the total distance traveled by the pendulum, we need to sum up the lengths of all the swings until it comes to rest. Since we know the first swing is 100 cm, and each swing is shorter than the previous one by a factor of 99/100, we have a geometric series.
The formula to calculate the sum of a geometric series is given by:
Sn = a₁ * (1 - r^n) / (1 - r),
where Sn is the sum of the first n terms, a₁ is the first term, and r is the common ratio.
We want to find the sum until the pendulum comes to rest, which is when the length becomes zero (0). So we need to find the number of swings until the length reaches zero.
To find n, we need to solve the equation:
a₁ * r^n = 0.
Since a₁ is a positive value (100 cm), we can multiply both sides by (1/r^n):
1 = 0.
There is no real solution to this equation, which means the pendulum will never come to rest.
Therefore, the total distance traveled by the pendulum will be infinite.