Acceleration of a particle moving on a circular path can be calculated using the formula:

\[a = \frac{{v^2}}{{r}}\]

where:
- \(a\) is the acceleration of the particle,
- \(v\) is the speed of the particle, and
- \(r\) is the radius of the circular path.

In this case, the speed of the particle is increasing at a rate of 2 meters per second squared. Therefore, we need to find the acceleration of the particle when its speed is 2 meters per second.

1. Finding the initial speed:
Given that the speed of the particle is 2 meters per second, we can conclude that the initial speed is also 2 meters per second.

2. Calculating the acceleration:
Using the formula for acceleration, we substitute the known values:
\[a = \frac{{(2 \, \text{m/s})^2}}{{1 \, \text{m}}} = \frac{{4 \, \text{m}^2/\text{s}^2}}{{1 \, \text{m}}} = 4 \, \text{m/s}^2\]

Therefore, the acceleration of the particle is 4 meters per second squared.

Explanation:
When a particle moves in a circular path, it experiences centripetal acceleration directed towards the center of the circle. This acceleration is perpendicular to the velocity vector at any point on the circular path and has a magnitude given by the formula mentioned above.

In this case, as the particle's speed is increasing, it means that there must be an additional force acting on the particle, causing its acceleration. This force is called tangential force, and it is responsible for the increase in speed.

The acceleration of the particle is directly proportional to the square of its speed, as shown in the formula. This is because the greater the speed, the larger the centripetal force required to keep the particle moving in a circular path of the same radius.

In summary, the particle will experience an acceleration of 4 meters per second squared as its speed increases at a rate of 2 meters per second squared.