A particle executes S.H.M. between `x = -A` and `x = + A`. The time taken for it to go from 0 to A/2 is `T_(1)` and to go from A/2 to A is `T_(2)`. Then

Options

A. `T_(1) lt T_(2)`
B. `T_(1) gt T_(2)`
C. `T_(1) = T_(2)`
D. `T_(1) = 2T_(2`

Explanation

To solve the problem, we need to find the relationship between the time taken for a particle executing Simple Harmonic Motion (SHM) to travel from 0 to A/2 (denoted as T1) and from A/2 to A (denoted as T2).

### Step-by-Step Solution:

1. **Understanding the SHM Equation**:
The position of a particle in SHM can be described by the equation:
\[ x = A \sin(\omega t) \]
where \( A \) is the amplitude, \( \omega \) is the angular frequency, and \( t \) is the time.

2. **Finding T1**:
For the particle to move from 0 to A/2, we set \( x = A/2 \):
\[ \frac{A}{2} = A \sin(\omega T_1) \]
Dividing both sides by \( A \) (assuming \( A \neq 0 \)):
\[ \frac{1}{2} = \sin(\omega T_1) \]
Taking the inverse sine:
\[ \omega T_1 = \sin^{-1}\left(\frac{1}{2}\right) = \frac{\pi}{6} \]
Thus, we can express \( T_1 \) as:
\[ T_1 = \frac{\pi}{6\omega} \]

3. **Finding T2**:
Now, for the particle to move from A/2 to A, we set \( x = A \):
\[ A = A \sin(\omega (T_1 + T_2)) \]
Dividing both sides by \( A \):
\[ 1 = \sin(\omega (T_1 + T_2)) \]
Taking the inverse sine:
\[ \omega (T_1 + T_2) = \sin^{-1}(1) = \frac{\pi}{2} \]
Thus, we can express \( T_1 + T_2 \) as:
\[ T_1 + T_2 = \frac{\pi}{2\omega} \]

4. **Finding T2**:
Now we can find \( T_2 \) by substituting \( T_1 \) into the equation:
\[ T_2 = (T_1 + T_2) - T_1 = \frac{\pi}{2\omega} - \frac{\pi}{6\omega} \]
To combine these fractions, we need a common denominator:
\[ T_2 = \frac{3\pi}{6\omega} - \frac{\pi}{6\omega} = \frac{2\pi}{6\omega} = \frac{\pi}{3\omega} \]

5. **Comparing T1 and T2**:
Now we have:
\[ T_1 = \frac{\pi}{6\omega} \quad \text{and} \quad T_2 = \frac{\pi}{3\omega} \]
We can see that:
\[ T_2 = 2T_1 \]
Therefore, \( T_2 \) is greater than \( T_1 \).

### Conclusion:
The relationship between the two times is:
\[ T_2 = 2T_1 \quad \text{or} \quad T_2 > T_1 \]