The formula [tex]$T=2 \pi \sqrt{\frac{L}{32}}$[/tex] relates the time, [tex]$T$[/tex], in seconds for a pendulum with the length, [tex]$L$[/tex], in feet, to make one full swing back and forth. What is the length of a pendulum that makes one full swing in 2.2 seconds? Use 3.14 for [tex]$\pi$[/tex].
A. 2 feet
B. 4 feet
C. 11 feet
D. 19 feet
Answer (2)
The length of the pendulum that makes a full swing in 2.2 seconds is approximately 4 feet. Thus, the correct answer is option B. This was calculated using the formula for the period of a pendulum and isolating the variable for length.
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Substitute the given values into the formula: 2.2 = 2 ( 3.14 ) 32 L .
Isolate the square root term: 2 ( 3.14 ) 2.2 = 32 L .
Square both sides of the equation: ( 2 ( 3.14 ) 2.2 ) 2 = 32 L .
Solve for L : L = 32 ( 2 ( 3.14 ) 2.2 ) 2 ≈ 4 feet.
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Explanation
Understanding the Problem We are given the formula T = 2 π 32 L which relates the time T in seconds for a pendulum with length L in feet to make one full swing back and forth. We are given that the time for one full swing is T = 2.2 seconds and π = 3.14 . We need to find the length L of the pendulum in feet.
Substituting the Values Substitute T = 2.2 and π = 3.14 into the formula: 2.2 = 2 ( 3.14 ) 32 L .
Isolating the Square Root Divide both sides by 2 ( 3.14 ) : 2 ( 3.14 ) 2.2 = 32 L .
Squaring Both Sides Square both sides: ( 2 ( 3.14 ) 2.2 ) 2 = 32 L .
Solving for L Multiply both sides by 32 to solve for L : L = 32 ( 2 ( 3.14 ) 2.2 ) 2 .
Calculating the Length Calculate the value of L : L = 32 ( 2 ( 3.14 ) 2.2 ) 2 ≈ 3.927 . Since we are looking for the closest answer from the given options, the length of the pendulum is approximately 4 feet.
Examples
Pendulums are used in clocks to keep time. Understanding the relationship between the length of a pendulum and its period (the time it takes for one full swing) is crucial in designing accurate timekeeping devices. For example, if you want to build a clock that ticks every second, you need to determine the precise length of the pendulum using the formula we just worked with. This ensures that the clock maintains accurate time by completing one full swing in a controlled amount of time.
