Question

T² vs l graph represents:

• A. A parabola
• B. A straight line passing through the origin
• C. A hyperbola
• D. A horizontal line

Hint:

Key points to remember:
- \( T^2 \propto l \) for simple pendulum
- Slope of graph gives \( g = \frac{4\pi^2}{\text{slope}} \)
- Graph must pass through origin (zero length $\Rightarrow$ zero period)
- This relationship holds for small oscillations only

Answer 1

The Correct Option is B

Step 1: Recall the physical relationship
For a simple pendulum, the period \( T \) is related to length \( l \) by: \[ T = 2\pi \sqrt{\frac{l}{g}} \] where \( g \) is acceleration due to gravity.
Step 2: Square both sides
\[ T^2 = \left(2\pi\right)^2 \frac{l}{g} \] \[ T^2 = \frac{4\pi^2}{g} l \]
Step 3: Recognize the equation form
This is of the form \( y = kx \), where: - \( y = T^2 \) - \( x = l \) - \( k = \frac{4\pi^2}{g} \) (constant)
Step 4: Determine graph characteristics
The equation \( T^2 = \left(\frac{4\pi^2}{g}\right)l \) represents: - A straight line - Passing through origin (0,0) - With slope \( \frac{4\pi^2}{g} \)
Step 5: Verify other options
(A) Parabola - Would require \( y \propto x^2 \)
(C) Hyperbola - Would require \( y \propto 1/x \)
(D) Horizontal line - Would mean \( T^2 \) is constant