Question

The displacement of a particle in simple harmonic motion is given by: \[ x = A_0 \cos \left( \frac{\pi}{2} t \right) \] The distance traveled by the particle in the interval between \( t = 2 \) and \( t = 5 \) seconds and its position at \( t = 5 \) second are:

• A. \( A_0 \) and mean position
• B. \( A_0 \) and extreme position
• C. \( 3 A_0 \) and mean position
• D. \( 3 A_0 \) and extreme position

Hint:

In simple harmonic motion, a particle oscillates symmetrically about the mean position with maximum displacement equal to the amplitude.

Answer 1

The Correct Option is C

Step 1: Understanding the SHM Equation

- The given displacement equation follows: \[ x = A_0 \cos \left( \frac{\pi}{2} t \right) \] - The amplitude of oscillation is \( A_0 \).
Step 2: Finding the Distance Traveled from \( t = 2 \) to \( t = 5 \)

- At \( t = 2 \): \[ x_2 = A_0 \cos \left( \frac{\pi}{2} \times 2 \right) = A_0 \cos (\pi) = -A_0 \] - At \( t = 5 \): \[ x_5 = A_0 \cos \left( \frac{\pi}{2} \times 5 \right) = A_0 \cos \left( \frac{5\pi}{2} \right) = 0 \] - The particle moves from \( -A_0 \) (extreme) → 0 → \( +A_0 \) (extreme) → 0 (mean position). - Total distance traveled = \( A_0 + 2 A_0 = 3 A_0 \).
Step 3: Finding the Final Position

- At \( t = 5 \), \( x_5 = 0 \), which means the particle is at the mean position.
Step 4: Conclusion

Since the total distance traveled is \( 3A_0 \) and the final position is the mean position, Option (3) is correct.