Solution:

Given:

- Force \( F = 6.4 \, \text{N} \)
- Extension \( x = 0.1 \, \text{m} \)
- Time period \( T = \frac{\pi}{4} \, \text{s} \)

1. Find the spring constant \( k \):

\[
k = \frac{F}{x} = \frac{6.4}{0.1} = 64 \, \text{N/m}
\]

2. Use the formula for the time period of a mass-spring system:

\[
T = 2\pi \sqrt{\frac{m}{k}}
\]

Substitute \( T = \frac{\pi}{4} \) and \( k = 64 \):

\[
\frac{\pi}{4} = 2\pi \sqrt{\frac{m}{64}}
\]

3. Solve for \( m \):

\[
\frac{1}{8} = \sqrt{\frac{m}{64}}
\]

\[
\frac{1}{64} = \frac{m}{64}
\]

\[
m = 1 \, \text{kg}
\]

Answer: \( m = 1 \, \text{kg} \)