Question:
Two particles \(A\) and \(B\) are executing simple harmonic motion with amplitudes \(10\,\text{cm}\) and \(20\,\text{cm}\) respectively. If the time periods of two particles \(A\) and \(B\) are \(8\,\text{s}\) and \(12\,\text{s}\) respectively, then the ratio of the times taken by the particles \(A\) and \(B\) to complete \(\dfrac18^{\text{th}}\) oscillation starting from their extreme positions is
Two particles \(A\) and \(B\) are executing simple harmonic motion with amplitudes \(10\,\text{cm}\) and \(20\,\text{cm}\) respectively. If the time periods of two particles \(A\) and \(B\) are \(8\,\text{s}\) and \(12\,\text{s}\) respectively, then the ratio of the times taken by the particles \(A\) and \(B\) to complete \(\dfrac18^{\text{th}}\) oscillation starting from their extreme positions is
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The time taken to complete any fraction of an oscillation is
\[
\boxed{t=\left(\text{Fraction of oscillation}\right)\times T.}
\]
It depends only on the time period \(T\) and is independent of the amplitude.
Updated On: Jul 18, 2026
- \(1:2\)
- \(1:1\)
- \(3:4\)
- \(2:3\)
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The Correct Option is D
Solution and Explanation
Step 1: Recall the relation between time and fraction of oscillation.
In simple harmonic motion, the time taken to complete a given fraction of one oscillation is directly proportional to the time period.
Thus, for completing
\[
\frac18
\]
of an oscillation,
\[
t=\frac{T}{8}.
\]
The amplitude does not affect the time taken.
Step 2: Calculate the required times. For particle \(A\), \[ t_A=\frac{8}{8}=1\,\text{s}. \] For particle \(B\), \[ t_B=\frac{12}{8}=\frac32\,\text{s}. \]
Step 3: Find the ratio. Therefore, \[ t_A:t_B = 1:\frac32 = 2:3. \] Hence, \[ \boxed{t_A:t_B=2:3.} \] Therefore, the correct option is \(\boxed{(D)}\).
Step 2: Calculate the required times. For particle \(A\), \[ t_A=\frac{8}{8}=1\,\text{s}. \] For particle \(B\), \[ t_B=\frac{12}{8}=\frac32\,\text{s}. \]
Step 3: Find the ratio. Therefore, \[ t_A:t_B = 1:\frac32 = 2:3. \] Hence, \[ \boxed{t_A:t_B=2:3.} \] Therefore, the correct option is \(\boxed{(D)}\).
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