Problem 71
Question
The period of a pendulum is the time it takes for the pendulum to make one full back-and-forth swing. The period of a pendulum depends on the length of the pendulum. The formula for the period \(P\), in seconds, is \(P=2 \pi \sqrt{\frac{l}{32}},\) where l is the length of the pendulum in feet. Find the length of a pendulum whose period is 4 seconds. Round your answer to 2 decimal places.
Step-by-Step Solution
Verified Answer
The length of the pendulum is 12.97 feet.
1Step 1: Write Down the Formula
The formula for the period of a pendulum is given by \( P = 2\pi \sqrt{\frac{l}{32}} \). We will use this formula to find the length, \( l \), when the period \( P \) is known.
2Step 2: Substitute the Known Period
We are given that the period \( P = 4 \) seconds. Substitute 4 for \( P \) in the formula: \( 4 = 2\pi \sqrt{\frac{l}{32}} \).
3Step 3: Solve for the Square Root Term
Divide both sides of the equation by \( 2\pi \) to isolate the square root term: \( \frac{4}{2\pi} = \sqrt{\frac{l}{32}} \).
4Step 4: Square Both Sides
Square both sides of the equation to eliminate the square root: \( \left( \frac{4}{2\pi} \right)^2 = \frac{l}{32} \).
5Step 5: Simplify and Solve for l
Calculate \( \left( \frac{4}{2\pi} \right)^2 \), which is approximately \( 0.40528 \). Now, solve for \( l \) by multiplying both sides by 32: \( l = 32 \times 0.40528 \).
6Step 6: Compute the Final Length
Perform the multiplication: \( l = 32 \times 0.40528 = 12.969 \). Round this to two decimal places to get \( l = 12.97 \).
Key Concepts
Pendulum LengthPhysics FormulaAlgebra ProblemSolving Equations
Pendulum Length
The length of a pendulum is a crucial factor affecting how the pendulum works. In physics, a pendulum is a weight or bob attached to the end of a long string or rod. When allowed to swing freely, the time it takes for the pendulum to complete one full cycle—swinging to and fro—is known as the pendulum's period. The pendulum length is directly related to the time of this period.
The length is not just the string or rod's distance; it’s the distance from the pivot point to the center of mass of the pendulum bob. With a longer length, the pendulum takes more time to swing back and forth compared to a shorter one. This is because a longer pendulum moves through a larger arc, which inherently requires more time.
The length is not just the string or rod's distance; it’s the distance from the pivot point to the center of mass of the pendulum bob. With a longer length, the pendulum takes more time to swing back and forth compared to a shorter one. This is because a longer pendulum moves through a larger arc, which inherently requires more time.
Physics Formula
The fundamental formula for determining a pendulum's period involves elements from mathematics and physics. It is given by the expression: \[ P = 2\pi \sqrt{\frac{l}{g}} \] Here, \( P \) stands for the period, \( l \) is the pendulum length, and \( g \) is the acceleration due to gravity. This equation shows how the period of a pendulum is related to its length and the gravitational force acting upon it.
For simplicity in many gravitational problems with a known constant, formulas are sometimes altered slightly. In our specific problem, the gravitational constant \( g \) is replaced by 32 feet per second squared, a value used in feet. This gives us a practical formula suitable for calculating the pendulum period specific to our scenario.
For simplicity in many gravitational problems with a known constant, formulas are sometimes altered slightly. In our specific problem, the gravitational constant \( g \) is replaced by 32 feet per second squared, a value used in feet. This gives us a practical formula suitable for calculating the pendulum period specific to our scenario.
Algebra Problem
Understanding this problem involves basic algebra, where you work with equations to find an unknown variable. In our exercise, the equation relates the pendulum's period to its length. We are given the period—4 seconds—and must determine the length \( l \).
Initially, we plug the known period into the pendulum formula before isolating the term with \( l \). This requires dividing and performing arithmetic operations to shift terms around in the equation effectively. The fundamental algebra skills used include substitution, manipulating equality through division and multiplication, and dealing with square roots.
Initially, we plug the known period into the pendulum formula before isolating the term with \( l \). This requires dividing and performing arithmetic operations to shift terms around in the equation effectively. The fundamental algebra skills used include substitution, manipulating equality through division and multiplication, and dealing with square roots.
Solving Equations
The process of solving equations is at the heart of finding the pendulum length in this scenario. Here's a breakdown of the key steps:
First, substitute the given period \( P = 4 \) into the formula \( P = 2\pi \sqrt{\frac{l}{32}} \). This substitution setups the problem with known values.
Next, to isolate the square root term, divide both sides by \( 2\pi \), resulting in: \[ \frac{4}{2\pi} = \sqrt{\frac{l}{32}} \]
To eliminate the square root, square both sides: \[ \left( \frac{4}{2\pi} \right)^2 = \frac{l}{32} \]
Then, solve for \( l \) by multiplying both sides by 32, leading to the calculation \( l = 32 \times 0.40528 \). Finally, perform the arithmetic, rounding as needed, to find the pendulum length: \( l = 12.97 \) feet. By following these systematic steps, the problem can be solved methodically and correctly.
First, substitute the given period \( P = 4 \) into the formula \( P = 2\pi \sqrt{\frac{l}{32}} \). This substitution setups the problem with known values.
Next, to isolate the square root term, divide both sides by \( 2\pi \), resulting in: \[ \frac{4}{2\pi} = \sqrt{\frac{l}{32}} \]
To eliminate the square root, square both sides: \[ \left( \frac{4}{2\pi} \right)^2 = \frac{l}{32} \]
Then, solve for \( l \) by multiplying both sides by 32, leading to the calculation \( l = 32 \times 0.40528 \). Finally, perform the arithmetic, rounding as needed, to find the pendulum length: \( l = 12.97 \) feet. By following these systematic steps, the problem can be solved methodically and correctly.
Other exercises in this chapter
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