Denver’s Altitude: How Does It Affect the Speed of a Pendulum’s Swing?

When considering experimental physics, the location of the experiment can sometimes have subtle but significant effects on the results. For instance, the altitude of a city like Denver can impact the speed of a pendulum's swing. Let's explore how and why.

Understanding the Impact of Altitude on Pendulum Swing Speed

The swing speed of a pendulum is directly related to the local gravitational acceleration, which is affected by the altitude above sea level. Denver, located in Colorado, sits between 1500 and 1800 meters above sea level. This altitude deviation changes the local gravitational acceleration, leading to a slight reduction in the pendulum's swing speed.

Calculating the Effect of Altitude on Gravitational Acceleration

The local gravitational acceleration g at any point is inversely proportional to the square of the distance from the Earth's center. Earth's average sea level is approximately 6,371,000 meters from the Earth's center. The ratio of gravitational acceleration at Denver's highest point to that at sea level can be calculated using the following formula:

x [6,371,000 / (6,371,000 1,800)]^2

This ratio helps us understand the impact on the pendulum's angular velocity ω, which is given by:

ω sqrt(g/L)

Where g is the local gravitational acceleration and L is the length of the pendulum's arm. The center of mass of the pendulum swings with a speed that is directly proportional to ω.

Measuring the Effect on Pendulum Speed

The speed of a pendulum can be expressed as:

A ω

Where A is the amplitude of the pendulum's swing. Therefore, the difference in speed between Denver and sea level can be calculated as the square root of the ratio of gravitational accelerations:

sqrt x sqrt [6,371,000 / (6,371,000 1,800)]^2

sqrt x 6,371,000 / (6,371,000 1,800)

sqrt x 0.999728

This means that the pendulum's speed is only about 0.0282% lower at Denver's highest point compared to sea level. In practical terms, this means that the pendulum would take about 24.4 seconds less per day to complete its swing compared to sea level.

Adjusting for the Altitude Effect

To correct for this small deviation in speed, one can make adjustments to the pendulum's setup. For example, reducing the length of the pendulum arm by a small percentage (0.0282%) or adding a small weight to the pendulum bob can help maintain the desired swing speed.

Conclusion

The altitude in Denver does indeed affect the speed of a pendulum's swing, but the effect is minimal. By understanding the underlying physics, we can make small adjustments to ensure accurate results in pendulum-based experiments. Whether you're a physicist or a curious science enthusiast, this knowledge provides an interesting insight into the practical considerations of conducting experiments at varying altitudes.

Keywords: pendulum swing, gravitational acceleration, Denver altitude, speed adjustment