Question

\(\cdot 17.41\) A clock based on a simple pendulum is situated outdoors in Anchorage, Alaska. The pendulum consists of a mass of 1.00 kg that is hanging from a thin brass rod that is \(2.000 \mathrm{~m}\) long. The clock is calibrated perfectly during a summer day with an average temperature of \(25.0^{\circ} \mathrm{C}\). During the winter, when the average temperature over one 24 -h period is \(-20.0^{\circ} \mathrm{C}\), find the time elapsed for that period according to the simple pendulum clock.

Short answer

Answer: The time elapsed according to the simple pendulum clock during a 24-hour period in winter is approximately 23.986 hours.

Step-by-step solution

Calculate the period of a simple pendulum in summer

Given the length of the pendulum, \(L = 2.000 m\) and \(g = 9.81 m/s^2\), we will calculate the period in summer: \(T_{summer} = 2\pi\sqrt{\frac{L}{g}}\) \(T_{summer} = 2\pi\sqrt{\frac{2.000}{9.81}}\) \(T_{summer} ≈ 2.839 s\)

Find the change in length due to the change in temperature

We are provided with information about the change in temperature from \(25.0^{\circ} C\) to \(-20.0^{\circ} C\): \(\Delta T = -45.0\) Kelvin. The linear expansion coefficient of brass, \(\alpha_{Brass} ≈ 19 \times 10^{-6} K^{-1}\). Now we calculate the change in length: \(\Delta L = \alpha_{Brass} \times L \times \Delta T\) \(\Delta L = 19 \times 10^{-6} \times 2.000 \times (-45)\) \(\Delta L ≈ -0.00171 m\)

Calculate the length of the pendulum in winter and the period in winter

Next, we calculate the length of the pendulum in winter: \(L_{winter} = L + \Delta L\) \(L_{winter} = 2.000 - 0.00171\) \(L_{winter} ≈ 1.99829 m\) Now, we will calculate the period of the pendulum in winter: \(T_{winter} = 2\pi\sqrt{\frac{L_{winter}}{g}}\) \(T_{winter} = 2\pi\sqrt{\frac{1.99829}{9.81}}\) \(T_{winter} ≈ 2.837 s\)

Determine the actual winter time according to the simple pendulum clock

Since the actual time the simple pendulum takes in winter is \(T_{winter} ≈ 2.837 s\), and during summer, it takes \(T_{summer} ≈ 2.839 s\) for one complete oscillation, we can set up the following equation: \(\frac{T_{winter}}{T_{summer}} = \frac{Actual~Winter~Time}{Someone~Provided~Time}\) \(\frac{2.837}{2.839} = \frac{Actual~Winter~Time}{24~hours}\) Now solve for the actual winter time: \(Actual~Winter~Time = \frac{2.837}{2.839} \times 24 hours\) \(Actual~Winter~Time ≈ 23.986 hours\) So, according to the simple pendulum clock, the time elapsed during a 24-hour period in winter is approximately \(23.986 hours\).

Key concepts

Thermal Expansion in Pendulums

Thermal expansion is a fundamental concept in physics where materials change in size and length as the temperature varies. This phenomenon is particularly significant in pendulums since they consist of materials such as metals that expand or contract with changes in temperature.

When it comes to a simple pendulum, which is a weight suspended from a fixed point by a rod or a string, the length of the pendulum is a critical factor determining its period—the time it takes to complete one full swing back and forth. As the temperature drops, the rod contracts, making the pendulum slightly shorter. Conversely, higher temperatures cause the rod to expand and the pendulum to lengthen.

Simple Pendulum Period

The period of a simple pendulum is the time it takes for the pendulum to swing from one side to the other and back again. It is described by the formula:
\( T = 2\pi\sqrt{\frac{L}{g}} \)
where \( T \) is the period, \( L \) is the length of the pendulum, and \( g \) is the acceleration due to gravity.

The period is directly proportional to the square root of the length, meaning that even small changes in the pendulum's length can have a considerable effect on the period. This relationship illustrates why temperature, through its impact on the pendulum's length via thermal expansion or contraction, plays a key role in the operation of pendulum-based timekeeping devices.

Temperature's Effect on Pendulum Length

Temperature changes can significantly affect the length of a pendulum because materials expand as they warm and contract as they cool down. Since the period of a pendulum is closely tied to its length, a pendulum will swing more quickly in cold temperatures and more slowly in warm ones. The linear expansion coefficient for a given material quantifies how much the material will expand or contract per unit length per degree change in temperature.

In practical terms, a simple pendulum clock calibrated during a warm summer day might 'lose' time during the cold winter, as the pendulum rod will contract, and the period becomes fractionally shorter. This change in period may seem insignificant over a single swing, but when accumulated over the hours, it can result in a notable time discrepancy and is an important consideration for precision timekeeping.