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].
Answer (1)
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 and multiply by 32 to solve for L : L = 32 ( 2 ( 3.14 ) 2.2 ) 2 .
Calculate the value of L to find the length of the pendulum: L ≈ 4 feet. Therefore, the answer is 4 f ee t .
Explanation
Understanding the Problem We are given the formula T = 2 π 32 L which relates the time T in seconds for a pendulum of length L in feet to make one full swing. We are given that T = 2.2 seconds and π = 3.14 . We want to find the length L of the pendulum.
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 .
Isolating L Multiply both sides by 32: L = 32 ( 2 ( 3.14 ) 2.2 ) 2 .
Calculating L Calculate the value of L : L = 32 ( 3.14 1.1 ) 2 ≈ 32 ( 0.3503 ) 2 ≈ 32 × 0.1227 ≈ 3.927 . Therefore, L ≈ 3.927 feet. Among the given choices, 4 feet is the closest to the calculated value.
Final Answer The length of the pendulum is approximately 3.927 feet, which is closest to 4 feet.
Examples
Pendulums are used in various applications, such as clocks and metronomes. Understanding the relationship between the length of a pendulum and its period (time for one full swing) is crucial in designing these devices. For example, if you want to build a clock that ticks every second, you need to determine the precise length of the pendulum to achieve this timing. This formula helps engineers and clockmakers accurately calculate the required pendulum length for specific timekeeping applications. The formula can also be used in seismometers to measure the frequency of earthquakes.
