Question

A certain pendulum clock (with a 12 -h dial) happens to gain 1 min/day. After setting the clock to the correct time, how long must one wait until it again indicates the correct time?

Short answer

The clock will indicate the correct time again after 1440 days.

Step-by-step solution

Identify the rate

The clock gains an extra 1 minute per day, which means in real 24-hour period, it shows 24 hours and 1 minute.

Calculate the total time until the clock indicates the correct time

To find out when the accumulated extra minutes add up to 24 hours, calculate how many minutes are in 24 hours and then divide by the rate at which the clock is gaining time. In 24 hours there are 1440 minutes (24 hours * 60 minutes). Divide 1440 by 1 to get the total time in days (1440 / 1 = 1440 days)

Conclusion

Therefore, the clock will again indicate the correct time after 1440 days.

Key concepts

Timekeeping Error

Timekeeping errors occur when a clock does not accurately show the passage of time. This could be because the clock is running too fast or too slow.
In the case of pendulum clocks, timekeeping errors can arise from errors in the oscillation period of the pendulum.
These errors can be caused by changes in the environment such as temperature or humidity, which affect the length or rigidity of the pendulum rod, leading to variations in motion.

• A pendulum clock gaining time means it ends up showing a time that is ahead of real time.
• The given problem states that the pendulum clock gains 1 minute every 24 hours, which is equivalent to running faster than real time by 1 minute each day.

Understanding timekeeping errors helps in evaluating how they influence long-term clock accuracy, which is essential for ensuring dependable timekeeping in everyday life.

Pendulum Motion

A pendulum's motion is key to how traditional clocks measure time. A pendulum consists of a weight, known as a bob, attached to the end of a rod or string.
When displaced, the pendulum swings back and forth in an arc due to gravity, a motion that is continuous and predictable.
The time it takes for one complete cycle of this swing is called the period.

• The period of a pendulum depends on the length of the pendulum and the gravitational force, not on the mass of the bob or the size of the swing.
• The formula for the period of a simple pendulum is given by: \( T = 2\pi \sqrt{\frac{L}{g}} \) where \( T \) is the period, \( L \) is the length of the pendulum, and \( g \) is the acceleration due to gravity.

This predictable motion allows pendulum clocks to keep time. However, if something affects the pendulum's length or gravity (like height change), it can lead to timekeeping errors, causing the clock to gain or lose time.

Clock Correction Methods

Correcting a pendulum clock that gains or loses time involves several practical methods:
Adjusting the pendulum's length is the most common solution. By lengthening the pendulum, the clock slows down, and by shortening it, the clock speeds up.

• A small adjustment in the length can have a significant effect on the period and thus the timekeeping.
• Many pendulum clocks have a small screw or nut at the pendulum's bottom which adjusts the length.
• Regulating the swing amplitude can also make minor corrections, though this is less common.

Other methods include:

• Using compensating devices that adjust for temperature changes, which would otherwise cause variations in the pendulum rod's length.
• Regular maintenance and calibration, which can help keep the clock accurate and account for mechanical wear or environmental changes.

These correction techniques help maintain the pendulum clock’s accuracy, thus minimizing the timekeeping error over extended periods.