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:
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:
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In simple harmonic motion, a particle oscillates symmetrically about the mean position with maximum displacement equal to the amplitude.
Updated On: Mar 6, 2026
- \( A_0 \) and mean position
- \( A_0 \) and extreme position
- \( 3 A_0 \) and mean position
- \( 3 A_0 \) and extreme position
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The Correct Option is C
Solution and Explanation
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.
- 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.
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