Questions: WWMFG3.1: Problem 3 (1 point) The length, L, of a pendulum varies directly with the square of its period, T, the time required for the pendulum to make one complete swing back and forth. The pendulum on a grandfather clock is 3.25 feet long and has a period of 2 seconds. (a) Express L as a function of T. Complete the table and graph your function in a suitable window. (b) How long is the Foucault pendulum in the Pantheon in Paris, which has a period of 17 seconds? Answer: (c) A hypnotist uses a gold pendant as a pendulum to mesmerize his clients. If the chain on the pendant is 11 inches long, what is the period of its swing? Answer: (d) In order to double the period of a pendulum, how must you vary its length?

WWMFG3.1: Problem 3
(1 point) The length, L, of a pendulum varies directly with the square of its period, T, the time required for the pendulum to make one complete swing back and forth. The pendulum on a grandfather clock is 3.25 feet long and has a period of 2 seconds.
(a) Express L as a function of T. Complete the table and graph your function in a suitable window.
(b) How long is the Foucault pendulum in the Pantheon in Paris, which has a period of 17 seconds? Answer:
(c) A hypnotist uses a gold pendant as a pendulum to mesmerize his clients. If the chain on the pendant is 11 inches long, what is the period of its swing? Answer:
(d) In order to double the period of a pendulum, how must you vary its length?

Transcript text: WWMFG3.1: Problem 3 (1 point) The length, $L$, of a pendulum varies directly with the square of its period, $T$, the time required for the pendulum to make one complete swing back and forth. The pendulum on a grandfather clock is 3.25 feet long and has a period of 2 seconds. (a) Express $L$ as a function of $T$. Complete the table and graph your function in a suitable window. (b) How long is the Foucault pendulum in the Pantheon in Paris, which has a period of 17 seconds? Answer: $\square$ (c) A hypnotist uses a gold pendant as a pendulum to mesmerize his clients. If the chain on the pendant is 11 inches long, what is the period of its swing? Answer: $\square$ (d) In order to double the period of a pendulum, how must you vary its length?

Solution

Solution Steps

Step 1: Express $ L $ as a function of $ T $

Given that the length $ L $ of a pendulum varies directly with the square of its period $ T $, we can write: \[ L = kT^2 \] where $ k $ is a constant of proportionality.

Step 2: Determine the constant $ k $

We know that when $ T = 2 $ seconds, $ L = 3.25 $ feet. Substituting these values into the equation: \[ 3.25 = k(2^2) \] \[ 3.25 = 4k \] \[ k = \frac{3.25}{4} = 0.8125 \]

Step 3: Write the function $ L(T) $

Now that we have $ k $, we can express $ L $ as a function of $ T $: \[ L = 0.8125T^2 \]

Final Answer

The function $ L(T) $ is: \[ L = 0.8125T^2 \]

Step 4: Calculate the length of the Foucault pendulum

For $ T = 17 $ seconds: \[ L = 0.8125(17^2) \] \[ L = 0.8125 \times 289 \] \[ L = 234.0625 \text{ feet} \]

Step 5: Calculate the period of the hypnotist's pendulum

For $ L = 11 $ inches (convert to feet: $ \frac{11}{12} $ feet): \[ \frac{11}{12} = 0.8125T^2 \] \[ T^2 = \frac{11}{12 \times 0.8125} \] \[ T^2 = \frac{11}{9.75} \] \[ T^2 = 1.1282 \] \[ T = \sqrt{1.1282} \] \[ T \approx 1.0622 \text{ seconds} \]

Final Answer

(a) The function $ L(T) $ is: \[ L = 0.8125T^2 \]

(b) The length of the Foucault pendulum is: \[ 234.0625 \text{ feet} \]

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