The curve between square of frequency of oscillation and length of the simple pendulum is

Options

A. Straight line
B. Parabolic
C. Ellipse
D. Hyperbola

Answer

To determine the relationship between the square of the frequency of oscillation and the length of a simple pendulum, we can follow these steps:

### Step 1: Understand the formula for frequency
The frequency \( f \) of a simple pendulum is given by the formula:
\[ f = \frac{1}{2\pi} \sqrt{\frac{g}{L}} \]
where:
- \( g \) is the acceleration due to gravity,
- \( L \) is the length of the pendulum.

### Step 2: Square the frequency
To find the square of the frequency, we square the equation:
\[ f^2 = \left(\frac{1}{2\pi} \sqrt{\frac{g}{L}}\right)^2 \]
This simplifies to:
\[ f^2 = \frac{g}{4\pi^2 L} \]

### Step 3: Rearrange the equation
We can rearrange this equation to express it in terms of \( L \):
\[ f^2 = \frac{g}{4\pi^2} \cdot \frac{1}{L} \]
This shows that \( f^2 \) is inversely proportional to \( L \).

### Step 4: Identify the type of curve
The equation \( f^2 = \frac{g}{4\pi^2} \cdot \frac{1}{L} \) indicates that as \( L \) increases, \( f^2 \) decreases. This relationship is characteristic of a hyperbola.

### Step 5: Graph the relationship
If we plot \( f^2 \) on the y-axis and \( L \) on the x-axis, the graph will show a hyperbolic curve, confirming that the relationship between the square of the frequency and the length of the pendulum is indeed a hyperbola.

### Conclusion
Thus, the curve between the square of the frequency of oscillation and the length of the simple pendulum is a hyperbola.

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