Application: Pendulum Clocks
Pendulums used in clocks, such as grandfather clocks, keep fairly accurate time for the following reason: When the length of a pendulum is large compared with the maximum arc of its swing, the time to complete one swing is independent of both the pendulum’s weight and the maximum displacement of the swing. When this condition is satisfied, the relationship between the time to complete one swing and the length of the pendulum is given by this formula:
length = 12.0 * g * [time/(2.0 * ð)]2
In this formula, ð accurate to four decimal places is equal to 3.1416, and g is the gravitational constant equal to 32.2 ft/sec2. When the time of a complete swing is given in seconds, the length of the pendulum is in feet. Using this formula, you’re to write a C++ program that calculates and displays the length of a pendulum needed to produce a swing that’s completed in 1 second. The length is to be displayed in inches.
Next, you apply the software development procedure to this problem.
To solve this problem, you can follow these steps using the software development procedure:
1. Identify the problem: The problem is to design a C++ program that calculates and displays the length of a pendulum needed to complete one swing in 1 second. The length should be displayed in inches.
2. Analyze the problem: From the given information, we know that the time to complete one swing (T) is 1 second. We need to use the given formula to calculate the length of the pendulum. The formula is:
length = 12.0 * g * [T/(2.0 * π)]^2
Here, g is the gravitational constant (32.2 ft/sec^2) and π is the mathematical constant (approximately 3.1416).
3. Design the solution: To solve the problem, we need to write a C++ program that performs the following steps:
a. Declare the necessary variables to store the gravitational constant, time, and length of the pendulum.
b. Calculate the length using the given formula.
c. Convert the length from feet to inches.
d. Display the length in inches.
4. Implement the solution: Here's an example C++ code that solves the problem:
```cpp
#include <iostream>
#include <cmath>
int main() {
// Declare variables
const double g = 32.2;
const double pi = 3.1416;
double time = 1.0;
double length;
// Calculate the length
length = 12.0 * g * pow(time / (2.0 * pi), 2);
// Convert length from feet to inches
length *= 12.0;
// Display the length in inches
std::cout << "The length of the pendulum needed to complete one swing in 1 second is: " << length << " inches" << std::endl;
return 0;
}
```
5. Test the solution: You can run the program and verify if it correctly calculates and displays the length of the pendulum for a 1-second swing in inches.
By following these steps, you can develop a C++ program that solves the given problem and get the desired outcome.