91Ó°ÊÓ

(a) What is the change \(\Delta T\) in the period of a simple penduIum when the acceleration of gravity \(g\) changes by \(\Delta g ?\) (Hint: The new period \(T+\Delta T\) is obtained by substituting \(g+\Delta g\) for \(g :\) $$\boldsymbol{T}+\Delta \boldsymbol{T}=2 \pi \sqrt{\frac{\boldsymbol{L}}{\boldsymbol{g}+\Delta \boldsymbol{g}}}$$ To obtain an approximate expression, expand the factor \((g+\Delta g)^{-1 / 2}\) using the binomial theorem (Appendix B) and keep only the first two terms: $$(g+\Delta g)^{-1 / 2}=g^{-1 / 2}-\frac{1}{2} g^{-3 / 2} \Delta g+\cdots$$ The other terms contain higher powers of \(\Delta g\) and are very small if \(\Delta g\) is small.) Express your result as the fractional change in period \(\Delta T / T\) in terms of the fractional change \(\Delta g / g .\) (b) A pendulum clock keeps correct time at a point where \(g=9.8000 \mathrm{m} / \mathrm{s}^{2},\) but is found to lose 4.0 \(\mathrm{s}\) each day at a higher elevation. Use the result of part (a) to find the approximate value of \(g\) at this new location.

Step by step solution

01

Understand the Problem

We need to find the change in the period of a pendulum when gravity changes by \( \Delta g \). The period of a pendulum is given by \( T = 2\pi \sqrt{\frac{L}{g}} \). The goal is to express \( \Delta T/T \) in terms of \( \Delta g/g \).

02

Modify the Period Formula

Substitute \( g + \Delta g \) into the formula: \( T + \Delta T = 2\pi \sqrt{\frac{L}{g + \Delta g}} \).

03

Simplify Using Binomial Theorem

Express \( (g + \Delta g)^{-1/2} \) using the binomial theorem: \( (g + \Delta g)^{-1/2} = g^{-1/2} - \frac{1}{2}g^{-3/2}\Delta g \).

04

Expand the New Period Formula

Substitute the expansion into the period equation: \[T + \Delta T = 2\pi \left(g^{-1/2} - \frac{1}{2}g^{-3/2}\Delta g \right)L^{1/2}\]This can be rewritten as: \[ T + \Delta T = T \left(1 - \frac{1}{2} \frac{\Delta g}{g} \right) \]

05

Find the Fractional Change in Period

Rearrange the expression from Step 4 to solve for \( \Delta T/T \):\[ \frac{\Delta T}{T} = -\frac{1}{2} \frac{\Delta g}{g} \]

06

Apply the Result to the Pendulum Clock Problem

Given the clock loses 4.0 s per day (or \( \frac{4}{86400} \) since a day has 86400 s), set \( \frac{\Delta T}{T} = \frac{-4}{86400} \). Substitute into this the result of part (a), \( -\frac{1}{2} \frac{\Delta g}{g} \). Solve for \( \frac{\Delta g}{g} \).

07

Calculate New Value of \( g \)

We find \( \frac{\Delta g}{g} = \frac{4}{86400 \times 2} \). Use initial \( g = 9.8000 \text{ m/s}^2 \) to find the new \( g \):\[ g' = g + \Delta g \] where \( \Delta g = \frac{4 \times 9.8000}{86400 \times 2} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Period of a Pendulum

A pendulum's period is the time it takes for a complete back-and-forth swing. This period is connected to the length of the pendulum (L) and the local acceleration due to gravity (g). The formula for the period (T) is given by:
\[ T = 2\pi \sqrt{\frac{L}{g}} \]
This equation shows that as the length increases, the period gets longer, while stronger gravity results in a shorter period. The period is independent of the mass of the pendulum and primarily dependent on the local gravitational force and its length. Understanding this formula is crucial to understanding how pendulums are used in various applications, such as timing devices and scientific experiments.

Gravity Variations

Gravity is not uniform everywhere on Earth. It can vary due to altitude, latitude, and even the local geological structure. When we move to a higher elevation, the gravitational acceleration ( g ) decreases. For pendulums, small changes in gravity will affect their oscillation period. This happens because, in the pendulum period formula, gravity is in the denominator. A small change in gravity can lead to a noticeable change in the period, particularly in precision instruments like pendulum clocks.
Accounting for gravity variations is important for applications such as clocks, surveying, and even for scientific research where precise timing is crucial.

Binomial Theorem

The binomial theorem is a powerful mathematical tool. It is used to expand expressions raised to a power. In our exercise, we use it to simplify calculations when gravity undergoes a small variation. For example, the term \((g + \Delta g)^{-1/2}\) can be approximated with the first two terms of its binomial expansion:
\[ (g + \Delta g)^{-1/2} \approx g^{-1/2} - \frac{1}{2} g^{-3/2} \Delta g \]
This simplification is accurate when \(\Delta g\) is much smaller than \(g\). This approximate expansion makes solving for changes in pendulum periods more straightforward as we can ignore very small terms that don't significantly affect the result.

Fractional Changes

Fractional changes provide a way to express how much a quantity changes in relation to its original value. In our exercise, the fractional change in the period of the pendulum is calculated using:
\[ \frac{\Delta T}{T} = -\frac{1}{2} \frac{\Delta g}{g} \]
This equation shows that for small changes in gravitational acceleration, the period's fractional change is directly proportional to the fractional change in gravity, albeit with a negative factor of a half. This concept is essential in practical applications, as it helps predict how pendulum behavior alters due to changing gravitational fields.

Pendulum Clock

Pendulum clocks are classical timekeeping devices that use the regular swing of a pendulum to mark the passage of time. However, an interesting challenge with these devices is that they can lose or gain time when moved to a location where gravity differs from their calibration point.

• A pendulum clock calibrated where \(g = 9.8 \, m/s^2\) might lose time at a location with weaker gravity.
• If the clock loses 4 seconds a day, this correlates to a measurable change in g associated with higher elevation or different geological conditions.

Using the knowledge of fractional changes, we can use:
\[ \frac{\Delta T}{T} = -\frac{1}{2} \frac{\Delta g}{g} \]
to estimate the new gravitational acceleration and correct the time discrepancy. This concept emphasizes the importance of accounting for environmental factors in precision timekeeping.