Question

For smaller angular displacement, the period of a simple pendulum depends on

• A. its amplitude
• B. its phase constant
• C. its energy
• D. the mass of the bob
• E. its length

Hint:

For small oscillations, remember \(T=2\pi\sqrt{\frac{l}{g}}\): depends only on length and gravity.

Answer 1

Answer: (E)

Step 1: Recall the formula for time period of a simple pendulum.
For small oscillations, the time period is: \[ T=2\pi\sqrt{\frac{l}{g}} \] where \(l\) is length and \(g\) is acceleration due to gravity.

Step 2: Identify variables affecting the time period.
From the formula, \(T\) depends only on \(l\) and \(g\), and not on mass or amplitude (for small angles).

Step 3: Analyze amplitude.
For small angular displacement, the motion is simple harmonic and the period is independent of amplitude.

Step 4: Analyze mass.
Mass of the bob does not appear in the formula, so it does not affect the time period.

Step 5: Analyze energy and phase.
Energy and phase constant are not present in the time period expression.

Step 6: Identify the correct dependency.
Only the length \(l\) directly affects the period.

Step 7: Final answer.
\[ \boxed{\text{its length}} \] which matches option \((5)\).