A simple pendulaum consists of a small mass suspended from a string that is fixed at its upper end and has negligible mass. The length of time, t second, for complete swing of a simple pendulum can be modeled by the equation `t = 2pisqrt(L/(32))`, where L is the length, in feet, of the string. If the time required for a complete swing of Pendulum 1 is triple the time required for a complete swing of Pendulum 2, the length of Pendulum 1's string is how many times the length of Pendulum 2's string?

Options

A. `1/3`
B. `3`
C. 6
D. 9

Answer

To solve the problem, we will follow these steps:

### Step 1: Write the formula for the time period of a simple pendulum.
The time period \( t \) of a simple pendulum is given by the formula:
\[ t = 2\pi \sqrt{\frac{L}{32}} \]
where \( L \) is the length of the string in feet.

### Step 2: Define variables for the two pendulums.
Let:
- \( t_1 \) = time period of Pendulum 1
- \( L_1 \) = length of the string of Pendulum 1
- \( t_2 \) = time period of Pendulum 2
- \( L_2 \) = length of the string of Pendulum 2

### Step 3: Write the equations for both pendulums using the formula.
For Pendulum 1:
\[ t_1 = 2\pi \sqrt{\frac{L_1}{32}} \]
For Pendulum 2:
\[ t_2 = 2\pi \sqrt{\frac{L_2}{32}} \]

### Step 4: Use the relationship between the time periods.
According to the problem, the time required for a complete swing of Pendulum 1 is triple that of Pendulum 2:
\[ t_1 = 3t_2 \]

### Step 5: Substitute the expressions for \( t_1 \) and \( t_2 \) into the equation.
Substituting the equations from Step 3 into the relationship from Step 4:
\[ 2\pi \sqrt{\frac{L_1}{32}} = 3 \times 2\pi \sqrt{\frac{L_2}{32}} \]

### Step 6: Cancel \( 2\pi \) from both sides.
This simplifies to:
\[ \sqrt{\frac{L_1}{32}} = 3 \sqrt{\frac{L_2}{32}} \]

### Step 7: Square both sides to eliminate the square root.
Squaring both sides gives:
\[ \frac{L_1}{32} = 9 \cdot \frac{L_2}{32} \]

### Step 8: Cancel \( 32 \) from both sides.
This simplifies to:
\[ L_1 = 9L_2 \]

### Step 9: Conclusion.
Thus, the length of Pendulum 1's string \( L_1 \) is 9 times the length of Pendulum 2's string \( L_2 \).

### Final Answer:
The length of Pendulum 1's string is **9 times** the length of Pendulum 2's string.

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