Problem 26 - Calculus Early Transcendentals: Pearson New International Edition

Question

The period of a simple pendulum of length \(L\) feet is given by \(T=2 \pi \sqrt{L / g}\) seconds. We assume that \(g\), the acceleration due to gravity on (or very near) the surface of the earth, is 32 feet per second per second. If the pendulum is that of a clock that keeps good time when \(L=4\) feet, how much time will the clock gain in 24 hours if the length of the pendulum is decreased to \(3.97\) feet?

Step-by-Step Solution

Answer

The clock will gain approximately 292.34 seconds in 24 hours.

Step 1: Finding the Initial Period

To find the initial period of the pendulum when the length is 4 feet, we use the formula for the period of a pendulum: \( T = 2 \pi \sqrt{\frac{L}{g}} \). Substituting \( L = 4 \) feet and \( g = 32 \) feet per second squared, we get: \[ T_1 = 2 \pi \sqrt{\frac{4}{32}} = 2 \pi \sqrt{\frac{1}{8}} = \pi \sqrt{\frac{1}{2}} = \pi \left( \frac{1}{\sqrt{2}} \right) = \frac{\pi}{\sqrt{2}} \] This is the time period, \( T_1 \), of the pendulum in seconds when \( L = 4 \) feet.

Step 2: Finding the Period with the New Length

Now, calculate the period with the adjusted length \( L = 3.97 \) feet. Using the same formula: \[ T_2 = 2 \pi \sqrt{\frac{3.97}{32}} = 2 \pi \sqrt{\frac{3.97}{32}} \approx 2 \pi \sqrt{0.12406} \approx 2 \pi (0.352187) \approx 2.213925 \] This gives the new period, \( T_2 \), in seconds.

Step 3: Calculate the Change in Period per Oscillation

To find out how much the period has changed per oscillation, subtract the original period from the new one: \[ \Delta T = T_2 - T_1 = 2.213925 - 2.221441 \approx -0.007516 \] This means the pendulum period has decreased by approximately 0.007516 seconds per oscillation.

Step 4: Determine Number of Oscillations in 24 Hours

To find out how many oscillations occur in 24 hours (86,400 seconds), divide the total seconds by the original period: \[ n = \frac{86400}{T_1} = \frac{86400}{2.221441} \approx 38910.39 \] So, there are approximately 38,910 oscillations in 24 hours.

Step 5: Calculate Total Time Gain in 24 Hours

The total time gained in 24 hours is the change in period per oscillation times the number of oscillations: \[ \text{Total Time Gain} = n \times \Delta T = 38910 \times -0.007516 \approx -292.34 \text{ seconds} \] This means that the clock will gain approximately 292.34 seconds over 24 hours when the pendulum length is reduced to 3.97 feet.

Key Concepts

Simple Pendulum

A simple pendulum is a fascinating mechanical system that consists of a weight suspended from a fixed point, allowing it to swing back and forth. This setup might remind you of the classic grandfather clock pendulums you might have seen. The primary components of a simple pendulum are:

• A mass (called the bob): This is the object that swings, usually a small, heavy object like a metal ball.
• A string or rod: Which is considered to have negligible mass and connects the bob to the pivot point.
• A pivot point: The fixed point from which the pendulum swings.

The motion of a simple pendulum is driven by gravity and can be described remarkably well by the equation for its period. This simplicity makes it an excellent tool for demonstrations in physics classes to explain concepts like harmonic motion.
The motion of a simple pendulum is periodic, performing the same swing over and over again in a constant time interval, assuming no air resistance or friction at the pivot.

Acceleration Due to Gravity

Acceleration due to gravity, often denoted as "g," is a crucial factor in pendulum motion. On the Earth's surface, this acceleration is approximately 32 feet per second squared, though this can vary slightly based on latitude and altitude. This value of gravity plays a significant role in the calculations for the time period of a pendulum's oscillation.
Gravity is what pulls the pendulum back towards its resting position after every swing. As the pendulum moves away from its equilibrium position, gravity tries to pull it back. This pull is responsible for the pendulum's acceleration.
Understanding how gravity impacts the pendulum's period helps in grasping broader physics concepts like harmonic motion and energy conservation. It also highlights why pendulums make excellent timekeeping devices since the gravitational pull remains consistent at a given location, ensuring uniform pendulum swings.

Oscillation Period

The oscillation period of a simple pendulum is the time it takes for one complete cycle of swing. This period is not influenced by the weight of the pendulum but rather by its length and the acceleration due to gravity. The formula for calculating the period,

• \[ T = 2 \pi \sqrt{\frac{L}{g}} \]

provides a direct relationship between these factors. Using this formula, you can determine how long it takes for a pendulum to complete one oscillation.
The period is a vital component in timekeeping. In the context of clocks, the pendulum's period must be precise to ensure the clock keeps accurate time.
In our previous exercise, we observed that by slightly altering the length of the pendulum from 4 feet to 3.97 feet, changes the oscillation period. This emphasizes how sensitive the pendulum period is to variations in length, which aligns with the principles outlined in harmonic oscillators. It’s essential for maintaining precision in a pendulum clock’s ability to measure time accurately.