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Chapter 5: Problem 17

A building that is subjected to shaking (caused, for example, by an earthquake) may collapse. Failure depends both on intensity and on duration of the shaking. If an intensity \(I_{1}\) causes a building to collapse in \(t_{1}\) seconds, then an intensity \(I_{2}\) will cause the collapse in \(t_{2}\) seconds, where $$ \frac{t_{1}}{t_{2}}=\left(\frac{I_{2}}{I_{1}}\right)^{2} $$ If a certain building collapses in 30 seconds at one intensity, how long would it take the building to collapse at triple that intensity?

Short Answer

Expert verified
The building will collapse in 3.33 seconds at triple the original intensity.

Step by step solution

01

Understand the Relationship Between Times and Intensities

The problem gives us a relationship between two times, \(t_1\) and \(t_2\), and their corresponding intensities, \(I_1\) and \(I_2\). This relationship is expressed as \( \frac{t_1}{t_2} = \left( \frac{I_2}{I_1} \right)^2 \). We need to apply this to find how long it takes the building to collapse at triple the initial intensity.
02

Assign Known Values

From the problem, we know \( t_1 = 30 \) seconds, \( I_2 = 3I_1 \), and we need to find \( t_2 \). This means the intensity \( I_2 \) is three times the initial intensity \( I_1 \).
03

Substitute and Simplify the Formula

Substitute the known values into the relationship: \( \frac{30}{t_2} = \left( \frac{3I_1}{I_1} \right)^2 \). This simplifies to \( \frac{30}{t_2} = 3^2 = 9 \).
04

Solve for \(t_2\)

Rearrange the equation to solve for \( t_2 \): \( t_2 = \frac{30}{9} = \frac{10}{3} \). This simplifies to \( t_2 = 3.33 \) seconds.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Intensity and Duration Relationship
Understanding the relationship between intensity and duration is crucial in solving problems involving dynamic forces, such as earthquakes. Intensity here refers to the strength or amplitude of the shaking. When a building is subjected to stronger forces, or higher intensity, it will take less time to reach the point of collapse compared to lower intensity. This is because the building absorbs energy at a faster rate as intensity increases.

In the exercise, we are provided with an equation: \[ \frac{t_1}{t_2} = \left( \frac{I_2}{I_1} \right)^2. \] This tells us that the ratio of times to failure, \(t_1\) and \(t_2\), is equal to the square of the ratio of intensities, \(I_2\) and \(I_1\). It means if you increase the intensity of the shaking, the duration until collapse becomes shorter at a rate that is the square of the change in intensity.

This mathematical model is an example of how engineering and physics can predict building responses in assessing how they might fare under different conditions.
Exponential Relationship
The relationship presented in the problem illustrates an exponential relationship between intensity and duration. An exponential formula is used when changes apply not by simple addition, but by multiplying. In this context, as you increase the intensity, the time until failure decreases at the square of that intensity increase.

Let鈥檚 break this down:
  • The equation \( \frac{t_1}{t_2} = \left( \frac{I_2}{I_1} \right)^2 \) describes how doubling the intensity more than halves the time to collapse.
  • In our exercise, tripling the intensity leads to the duration reducing by a factor of nine, calculated by squaring three (\(3^2 = 9\)).
Understanding this exponential relationship helps engineers and scientists predict how rapidly a building could succumb to varying earthquake intensities, enabling more informed safety measures and building codes.
Problem Solving Steps
Here's a straightforward guide to solving problems like this one, involving intensity and duration of shaking. First, identify the given values and what you need to find out. In this example, you are provided with \(t_1 = 30\) seconds and \(I_2 = 3I_1\). You need to determine \(t_2\), the time at tripled intensity.

Next, substitute the known values into the relationship formula. Replace \(I_2\) with \(3I_1\) in the equation \(\frac{t_1}{t_2} = \left(\frac{I_2}{I_1}\right)^2\). This will provide the equation: \[ \frac{30}{t_2} = (3)^2 = 9. \] Now solve for \(t_2\) by rearranging the equation: \[ t_2 = \frac{30}{9} = \frac{10}{3} = 3.33 \text{ seconds}. \] This step-by-step approach is crucial in determining how changes in one variable affect another, helping you to accurately model the problem and anticipate outcomes effectively.

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Most popular questions from this chapter

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