Problem 50 - Algebra 2

Question

Physics 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 a pendulum has a length of 100 \(\mathrm{cm}\) and the return swing is 99 \(\mathrm{cm} .\) a. On which swing will the arc first have a length less than 50 \(\mathrm{cm} ?\) b. Find the total distance traveled by the pendulum until it comes to rest.

Step-by-Step Solution

Answer

a. The pendulum's arc will first have a length less than 50 cm on the 70th swing. b. The total distance traveled by the pendulum until it comes to rest is approximately 10000 cm.

Step 1: Understand the Problem

The pendulum swing lengths form a geometric sequence where each swing is 99% of the previous swing. Here, the rate or common ratio is 0.99, and the first term is 100 cm.

Step 2: Calculate Which Swing Will Have the Length Less Than 50 cm

Set up the inequality \(100*(0.99)^n < 50\) and solve for \(n\). By taking the natural log and simplifying, one gets \(n > log_{0.99}(0.5)\). Since \(n\) must be an integer, it is rounded up to give the result.

Step 3: Calculate the Total Distance Traveled

The total distance traveled will be the sum of an infinite geometric series, given by the formula \(S = a / (1 - r) \), where \(a\) is the first term and \(r\) is the common ratio. Substituting \(a=100\) and \(r=0.99\), the total distance is calculated.

Key Concepts

Pendulum Swing

A pendulum is a simple system in physics where a weight, or bob, is hung from a fixed point and can swing back and forth. In real life situations, friction and air resistance act on the pendulum, meaning that each swing is slightly shorter than the last. The lengths of these swings decrease progressively, not moving forever, but getting smaller until they come to a stop.

Imagine a pendulum set in motion with an initial swing length. This initial swing will be longer compared to the subsequent swings due to energy lost in overcoming friction and air resistance. These swings can be expressed as a geometric sequence highlighting how physics makes use of mathematical relationships to predict outcomes.

Common Ratio

In geometric sequences, each term after the first is found by multiplying the previous one by a fixed, non-zero number. This number is called the common ratio. In the case of our pendulum, the common ratio is 0.99.

This means each swing is 99% the length of the previous swing. For example, if the first swing is 100 cm, the next swing would be 99 cm, followed by 98.01 cm, and so forth.

• Common ratio less than 1 implies a decreasing sequence.
• Common ratio greater than 1 implies a growing sequence.

Understanding the common ratio helps in calculating future terms and determining long-term behavior of the sequence.

Infinite Series

The notion of an infinite series comes into play when we need to find the total distance traveled by the pendulum swing over time. As the swings continue to decrease in length, they form an infinite series because we are adding an infinite number of decreasing terms.

The sum of this series is calculated using the formula:
\[S = \frac{a}{1 - r}\] where \(a\) is the first term and \(r\) is the common ratio.

• For the pendulum, \(a = 100\) cm and \(r = 0.99\).
• This helps determine the total distance of all swings until they are imperceptibly small.

This concept shows how even infinite additions can result in finite sums due to the properties of geometric sequences.

Natural Log

Natural logarithms make complex mathematical operations manageable by allowing us to solve equations involving exponential growth or decay. When determining which swing of the pendulum falls below a specific length, like 50 cm, we use natural logarithms.

For the pendulum problem, the equation is set up as:
\[100 \times (0.99)^n < 50\]
To solve for \(n\), we take the natural logarithm of both sides enabling us to rearrange the equation into:
\[n > \ln(0.5) / \ln(0.99)\]

• This tells us that the natural logarithm is a useful tool for handling exponential functions.
• It's crucial in computing exact moments when conditions are met in a geometric sequence.

Calculating natural logs is fundamental in many aspects of mathematics and physics due to their ability to unravel exponential equations.