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.
a.) To find the swing on which the length first becomes less than 50 cm, we can set up a geometric sequence using the given information. We know that the first swing has an arc length of 100 cm and the return swing has an arc length of 99 cm. Since the lengths of the swings form a geometric sequence, we can use the formula for geometric sequences to find the nth term.
The formula for the nth term of a geometric sequence is given by:
an = a * r^(n-1)
where an is the nth term, a is the first term, r is the common ratio, and n is the term number.
In this case, the first term (a) is 100 cm and the return swing (99 cm) is the second term (n = 2). We can set up the equation as follows:
99 = 100 * r^(2-1)
Simplifying the equation, we get:
99 = 100 * r
Dividing both sides by 100, we get:
r = 99/100
Now we can find the term number (n) for when the length first becomes less than 50 cm. We set up the equation:
50 = 100 * (99/100)^(n-1)
Dividing both sides by 100, we get:
0.5 = (99/100)^(n-1)
To solve for n, we can take the logarithm of both sides, using the base of the common ratio (99/100):
log base (99/100) of 0.5 = n - 1
Using logarithmic properties, we can rewrite the equation as:
log base (99/100) of 0.5 + 1 = n
Using a calculator or math software to evaluate the logarithm, we find that n is approximately 68.28.
Therefore, on approximately the 69th swing (n = 69), the length of the swing will be less than 50 cm.
b.) To find the total distance traveled by the pendulum until it comes to rest, we need to sum up the lengths of all the swings in the geometric sequence.
The sum of a geometric series can be found using the formula:
Sn = a * (1 - r^n) / (1 - r)
where Sn is the sum of the first n terms, a is the first term, r is the common ratio, and n is the number of terms.
In this case, a is 100 cm, r is 99/100, and we want to find the sum until the length becomes less than 50 cm (approximately on the 69th swing). So n = 69.
Using the formula, we can calculate the sum:
S69 = 100 * (1 - (99/100)^69) / (1 - 99/100)
Using a calculator or math software, we find that the sum S69 is approximately 199.3 cm.
Therefore, the total distance traveled by the pendulum until it comes to rest is approximately 199.3 cm.