A particle is subjected to two simple harmonic motion in the same direction having equal amplitudes and equal frequency. If the resultant amplitude is equal to the amplitude of the individual motions. Find the phase difference between the individual motions.

Explanation

To solve the problem, we need to find the phase difference between two simple harmonic motions (SHMs) that have equal amplitudes and frequencies, given that the resultant amplitude is equal to the amplitude of the individual motions.

### Step-by-Step Solution:

1. **Understand the Problem**:
We have two simple harmonic motions (SHMs) with equal amplitudes \( A \) and equal frequencies. The resultant amplitude of these two SHMs is given to be equal to the amplitude of the individual motions.

2. **Write the Formula for Resultant Amplitude**:
The resultant amplitude \( R \) of two SHMs can be expressed as:
\[ R = \sqrt{A_1^2 + A_2^2 + 2A_1A_2 \cos \phi} \]
Since both amplitudes are equal, we can denote them as \( A_1 = A \) and \( A_2 = A \). Therefore, the formula simplifies to:
\[ R = \sqrt{A^2 + A^2 + 2A \cdot A \cos \phi} \]

3. **Substitute the Resultant Amplitude**:
According to the problem, the resultant amplitude \( R \) is equal to the individual amplitude \( A \):
\[ A = \sqrt{A^2 + A^2 + 2A^2 \cos \phi} \]

4. **Square Both Sides**:
To eliminate the square root, we square both sides:
\[ A^2 = A^2 + A^2 + 2A^2 \cos \phi \]
This simplifies to:
\[ A^2 = 2A^2 + 2A^2 \cos \phi \]

5. **Rearrange the Equation**:
Rearranging gives:
\[ 0 = A^2 + 2A^2 \cos \phi \]
Factoring out \( A^2 \):
\[ 0 = A^2(1 + 2 \cos \phi) \]

6. **Solve for Cosine**:
Since \( A^2 \) is not zero, we can divide by \( A^2 \):
\[ 1 + 2 \cos \phi = 0 \]
This leads to:
\[ 2 \cos \phi = -1 \]
Thus:
\[ \cos \phi = -\frac{1}{2} \]

7. **Find the Phase Difference**:
The angle \( \phi \) for which \( \cos \phi = -\frac{1}{2} \) corresponds to:
\[ \phi = 120^\circ \quad \text{or} \quad \phi = 240^\circ \]
However, since we are looking for the phase difference, we can take \( \phi = 120^\circ \).

### Final Answer:
The phase difference between the individual motions is \( 120^\circ \).