The formula $\sqrt{\frac{ L }{32}}$ gives the time it takes in seconds, $T$, for a pendulum to make one full swing back and forth, where $L$ is the length of the pendulum, in feet. To the nearest foot, what is the length of a pendulum that makes one full swing in 1.9 s? Use 3.14 for $\pi$.
____ ft
Answer (2)
The length of the pendulum that makes one full swing in 1.9 seconds is approximately 116 feet, after applying the given formula and rounding to the nearest foot.
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Substitute the given time T = 1.9 into the formula T = 32 L .
Square both sides of the equation to get 1. 9 2 = 32 L , which simplifies to 3.61 = 32 L .
Multiply both sides by 32 to solve for L : L = 32 × 3.61 = 115.52 .
Round L to the nearest foot: L ≈ 116 ft.
Explanation
Understanding the Problem We are given the formula $T =
\sqrt{\frac{L}{32}} , w hi c h re l a t es t h e t im e T in seco n d s f or a p e n d u l u m t o mak eo n e f u ll s w in g t o i t s l e n g t h L in f ee t . W e a re g i v e n t ha tt h e t im e f oro n e f u ll s w in g i s 1.9 seco n d s , so T = 1.9 . W e n ee d t o f in d t h e l e n g t h L$ of the pendulum to the nearest foot.
Substituting the Value of T Substitute T = 1.9 into the formula:
1.9 = 32 L
To solve for L , we first square both sides of the equation:
( 1.9 ) 2 = 32 L
3.61 = 32 L
Solving for L Now, multiply both sides of the equation by 32 to isolate L :
L = 32 × 3.61
L = 115.52
Rounding to the Nearest Foot We are asked to round the value of L to the nearest foot. Since the decimal part of 115.52 is 0.52, which is greater than or equal to 0.5, we round up to the nearest whole number.
L ≈ 116
Therefore, the length of the pendulum is approximately 116 feet.
Final Answer The length of the pendulum that makes one full swing in 1.9 seconds, rounded to the nearest foot, is 116 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 (the time it takes for one full swing) is crucial in designing these devices. For example, if you want to build a clock with a pendulum that swings once every second, you can use the formula to determine the required length of the pendulum. This ensures accurate timekeeping in the clock's mechanism.
