Time `S.H.M.` of equal amplitude `a` and equal time period in the same direction combine. The first is `60^(@)` ahead of the second and second is `60^(@)` ahead of the third `S.H.M`. The amplitude of the resultant oscillation is:

Options

A. a
B. `2a`
C. 0
D. `4a`

Explanation

To solve the problem of finding the amplitude of the resultant oscillation when three simple harmonic motions (SHMs) combine, we can follow these steps:

### Step 1: Understand the Problem
We have three SHMs with the same amplitude `a` and the same time period. The phase differences between them are given as follows:
- The first SHM is `60°` ahead of the second.
- The second SHM is `60°` ahead of the third.

### Step 2: Represent the SHMs
Let’s denote the three SHMs as:
- \( S_1 = a \sin(\omega t + 0°) \) (first SHM)
- \( S_2 = a \sin(\omega t + 60°) \) (second SHM)
- \( S_3 = a \sin(\omega t + 120°) \) (third SHM)

### Step 3: Calculate the Resultant of the First Two SHMs
To find the resultant of the first two SHMs, we can use the formula for the resultant amplitude of two SHMs:
\[ R_{12} = \sqrt{A_1^2 + A_2^2 + 2A_1A_2 \cos(\phi)} \]
where \( A_1 = A_2 = a \) and \( \phi = 60° \).

Substituting the values:
\[ R_{12} = \sqrt{a^2 + a^2 + 2a^2 \cos(60°)} \]
Since \( \cos(60°) = \frac{1}{2} \):
\[ R_{12} = \sqrt{a^2 + a^2 + 2a^2 \cdot \frac{1}{2}} = \sqrt{a^2 + a^2 + a^2} = \sqrt{3a^2} = a\sqrt{3} \]

### Step 4: Calculate the Resultant of \( R_{12} \) and \( S_3 \)
Now we need to find the resultant of \( R_{12} \) and the third SHM \( S_3 \):
- The phase difference between \( R_{12} \) and \( S_3 \) is \( 120° \) (since \( R_{12} \) is effectively at \( 60° \) and \( S_3 \) is at \( 120° \)).

Using the same formula for the resultant:
\[ R = \sqrt{R_{12}^2 + A_3^2 + 2R_{12}A_3 \cos(120°)} \]
Substituting \( R_{12} = a\sqrt{3} \) and \( A_3 = a \):
\[ R = \sqrt{(a\sqrt{3})^2 + a^2 + 2(a\sqrt{3})(a) \cos(120°)} \]
Since \( \cos(120°) = -\frac{1}{2} \):
\[ R = \sqrt{3a^2 + a^2 - (a^2\sqrt{3})} = \sqrt{3a^2 + a^2 - \sqrt{3}a^2} = \sqrt{(4 - \sqrt{3})a^2} \]

### Step 5: Final Result
The amplitude of the resultant oscillation is:
\[ R = a\sqrt{4 - \sqrt{3}} \]

### Conclusion
Thus, the amplitude of the resultant oscillation is \( a\sqrt{4 - \sqrt{3}} \).