Problem 17
Question
The period (in seconds) for a pendulum of length \(L\) in. to complete one oscillation is equal to \(P=0.324 \sqrt{L} .\) Find the rate of change of the period with respect to length when the length is 9.00 in.
Step-by-Step Solution
Verified Answer
The rate of change of the period with respect to the length when the length is 9.00 in. is 0.054 sec/in.
1Step 1: Write the formula for period
Start by stating the formula for the period of a pendulum: \( P = 0.324 \sqrt{L} \).
2Step 2: Differentiate with respect to length
To find the rate of change of period with respect to length, differentiate the period function with respect to \( L \). Use the chain rule: \( \frac{dP}{dL} = 0.324 \frac{1}{2}\sqrt{L}^{-\frac{1}{2}} = 0.324 \cdot \frac{1}{2\sqrt{L}} = \frac{0.324}{2\sqrt{L}} \).
3Step 3: Plug in the length value
Substitute \( L = 9.00 \) in. into the derivative to find the rate of change at that specific length: \( \frac{dP}{dL} = \frac{0.324}{2\sqrt{9.00}} \).
4Step 4: Simplify the expression
Simplify the expression, remembering that \( \sqrt{9.00} = 3 \): \( \frac{dP}{dL} = \frac{0.324}{2 \times 3} = \frac{0.324}{6} \).
5Step 5: Calculate the rate of change
Perform the division to find the rate of change of the period with respect to the length: \( \frac{dP}{dL} = \frac{0.324}{6} = 0.054 \) sec/in.
Key Concepts
Differential CalculusChain RulePendulum Period Formula
Differential Calculus
Differential calculus is a fundamental mathematical tool that deals with changes and rates of change. In the context of physical problems, like the swinging of a pendulum, understanding how to use differential calculus is key to predicting and understanding dynamic systems. For instance, knowing how the period of a pendulum changes as its length changes is crucial for precise engineering applications.
When we talk about finding the rate of change, we are typically referring to taking a derivative. A derivative of a function gives us the rate at which the function's value is changing at any given point. When applied to the formula for the pendulum's period, taking the derivative with respect to the pendulum's length, L, allows us to find how sensitive the period is to slight changes in length. This sensitivity is crucial for fine-tuning in precision timekeeping or synchronized mechanisms in mechanical engineering.
When we talk about finding the rate of change, we are typically referring to taking a derivative. A derivative of a function gives us the rate at which the function's value is changing at any given point. When applied to the formula for the pendulum's period, taking the derivative with respect to the pendulum's length, L, allows us to find how sensitive the period is to slight changes in length. This sensitivity is crucial for fine-tuning in precision timekeeping or synchronized mechanisms in mechanical engineering.
Chain Rule
The chain rule is a fundamental theorem in differential calculus that allows us to differentiate composite functions. A composite function is a function made up of inner and outer functions, analogous to layers of an onion. To differentiate these, we start with the outer layer and work our way inward, multiplying the derivatives of each layer as we go.
For example, in our pendulum period problem, the period formula has a square root function, which we treat as the outer function, and the pendulum length, L, as the inner function. According to the chain rule, we first differentiate the square root (outer function) and multiply it by the derivative of L (inner function), which results in the derivative of the period with respect to the length. This method allows us to systematically tackle more complex rates of change such as these in various scientific and engineering problems.
For example, in our pendulum period problem, the period formula has a square root function, which we treat as the outer function, and the pendulum length, L, as the inner function. According to the chain rule, we first differentiate the square root (outer function) and multiply it by the derivative of L (inner function), which results in the derivative of the period with respect to the length. This method allows us to systematically tackle more complex rates of change such as these in various scientific and engineering problems.
Pendulum Period Formula
The pendulum period formula, given by \( P = 0.324 \sqrt{L} \), is a simple equation that relates the time for one full swing of a pendulum (period) to the length of the pendulum. According to physics, the period of a simple pendulum is proportional to the square root of its length, assuming we are ignoring air resistance and other non-ideal factors.
In practical applications, the length of a pendulum can be adjusted to make the pendulum swing with a specific rhythm. This is especially useful in timekeeping, such as in grandfather clocks. Using the pendulum period formula, clockmakers can easily calculate the necessary length for a pendulum to achieve a precise one-second period, which is foundational in maintaining consistent time measurement. Our exercise focuses on understanding how small changes in the length of the pendulum affect the period, which is paramount for fine adjustments and calibrations.
In practical applications, the length of a pendulum can be adjusted to make the pendulum swing with a specific rhythm. This is especially useful in timekeeping, such as in grandfather clocks. Using the pendulum period formula, clockmakers can easily calculate the necessary length for a pendulum to achieve a precise one-second period, which is foundational in maintaining consistent time measurement. Our exercise focuses on understanding how small changes in the length of the pendulum affect the period, which is paramount for fine adjustments and calibrations.
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