Magnitude of Net Acceleration in Circular Motion

In order to calculate the magnitude of net acceleration of a particle moving on a circular path, we need to consider two components of acceleration: tangential acceleration and centripetal acceleration. The tangential acceleration is responsible for the change in the particle's speed, while the centripetal acceleration keeps the particle moving in a circular path.

1. Tangential Acceleration:
The tangential acceleration can be calculated using the formula:
a_t = dv/dt
where a_t is the tangential acceleration, dv is the change in velocity, and dt is the change in time.

Given that the particle's speed is increasing at a rate of 2 m/s, we can substitute dv = 2 m/s and dt = 1 s into the formula to find the tangential acceleration:
a_t = 2 m/s / 1 s = 2 m/s^2

2. Centripetal Acceleration:
The centripetal acceleration can be calculated using the formula:
a_c = v^2 / r
where a_c is the centripetal acceleration, v is the velocity of the particle, and r is the radius of the circular path.

Given that the radius of the circular path is 10 m and the speed of the particle is 5 m/s, we can substitute these values into the formula to find the centripetal acceleration:
a_c = (5 m/s)^2 / 10 m = 2.5 m/s^2

3. Net Acceleration:
The net acceleration is the vector sum of the tangential acceleration and the centripetal acceleration. Since these two accelerations act at right angles to each other, we can use the Pythagorean theorem to find the magnitude of the net acceleration:
a_net = sqrt(a_t^2 + a_c^2)

Substituting the values we calculated earlier, we find:
a_net = sqrt((2 m/s^2)^2 + (2.5 m/s^2)^2)
= sqrt(4 m^2/s^4 + 6.25 m^2/s^4)
= sqrt(10.25 m^2/s^4)
≈ 3.2 m/s^2

Therefore, at the instant when the particle's speed is 5 m/s and increasing at a rate of 2 m/s, the magnitude of the net acceleration is approximately 3.2 m/s^2. This net acceleration combines both the tangential acceleration responsible for changing the speed and the centripetal acceleration responsible for keeping the particle in its circular path.