The magnitude of the centripetal acceleration of the stone while in circular motion is 31.7 m/s².
First, we need to find the initial velocity of the rock before it breaks free. We can use the formula for the velocity of a point on the circumference of a circle:
V = ωr
Where V is the velocity, ω is the angular velocity, and r is the radius. Since the rock strikes the ground after traveling a horizontal distance of 23.52 m, we can set up the following equation:
23.52 = ω(5.2/2)
Simplifying, we get:
ω = (23.52 × 2)/5.2 = 9 m/s
Now, we can find the time it takes for the rock to fall to the ground using the formula:
t = √(2h/g)
Where t is the time, h is the height, and g is the acceleration due to gravity. Plugging in the values:
t = √(2 × 8.56/9.8) = 1.5 s
The centripetal acceleration is given by the formula:
a = v² / r
Where a is the acceleration, v is the velocity, and r is the radius. Plugging in the values:
a = 9² / (5.2/2) = 31.7 m/s²
So, the magnitude of the centripetal acceleration of the stone while in circular motion is 31.7 m/s².