A particle is executing `SHM`. Find the positions of the particle where its speed is `8 cm//s`, if maximum magnitudes of its velocity and accleration are `10 cm//s` and `50 cm//s` respectively.

Explanation

To solve the problem, we need to find the positions of a particle executing simple harmonic motion (SHM) where its speed is 8 cm/s, given that the maximum velocity is 10 cm/s and the maximum acceleration is 50 cm/s².

### Step-by-Step Solution:

1. **Identify the given values:**
- Maximum velocity, \( V_{max} = 10 \, \text{cm/s} \)
- Maximum acceleration, \( A_{max} = 50 \, \text{cm/s}^2 \)
- Speed at which we want to find the position, \( v = 8 \, \text{cm/s} \)

2. **Relate maximum velocity to amplitude and angular frequency:**
The maximum velocity in SHM is given by:
\[ V_{max} = A \omega \]
where \( A \) is the amplitude and \( \omega \) is the angular frequency.

3. **Relate maximum acceleration to amplitude and angular frequency:**
The maximum acceleration in SHM is given by:
\[ A_{max} = A \omega^2 \]

4. **Set up equations from the given maximum values:**
From the maximum velocity:
\[ A \omega = 10 \quad \text{(Equation 1)} \]
From the maximum acceleration:
\[ A \omega^2 = 50 \quad \text{(Equation 2)} \]

5. **Divide Equation 2 by Equation 1 to find \( \omega \):**
\[ \frac{A \omega^2}{A \omega} = \frac{50}{10} \]
Simplifying gives:
\[ \omega = 5 \, \text{rad/s} \]

6. **Substitute \( \omega \) back into Equation 1 to find \( A \):**
\[ A \cdot 5 = 10 \implies A = \frac{10}{5} = 2 \, \text{cm} \]

7. **Use the velocity formula to find positions where speed is 8 cm/s:**
The velocity in SHM is given by:
\[ v = \omega \sqrt{A^2 - x^2} \]
Substituting the known values:
\[ 8 = 5 \sqrt{2^2 - x^2} \]

8. **Solve for \( x \):**
\[ \frac{8}{5} = \sqrt{4 - x^2} \]
Squaring both sides:
\[ \left(\frac{8}{5}\right)^2 = 4 - x^2 \]
\[ \frac{64}{25} = 4 - x^2 \]
Rearranging gives:
\[ x^2 = 4 - \frac{64}{25} \]
Converting 4 to a fraction:
\[ x^2 = \frac{100}{25} - \frac{64}{25} = \frac{36}{25} \]
Taking the square root:
\[ x = \pm \frac{6}{5} \, \text{cm} \]

9. **Final answer:**
The positions of the particle where its speed is 8 cm/s are:
\[ x = \frac{6}{5} \, \text{cm} \quad \text{and} \quad x = -\frac{6}{5} \, \text{cm} \]