Question: In a simple pendulum experiment, the bob takes time t1 to move from its extreme position A to some intermediate point C, and t2 to move from C to the mean position O. Prove that the time period T of the pendulum is \( 4 \bigl(t_{1} + t_{2}\bigr). \)
Given: The bob moves from the extreme position A to an intermediate point C in a time t1, and from C to the mean position O in t2.
To Find: The time period T of the pendulum.
Key Idea: For a simple pendulum undergoing SHM (Simple Harmonic Motion), it takes one-fourth of the total time period (T) to go from an extreme position (A) to the mean position (O). Hence,
The total time from A to O can also be written as the sum of the times from A to C and C to O:
Rearranging this equation to solve for T gives:
Answer: The time period of the simple pendulum is \( T = 4(t_{1} + t_{2}) \).