A certain string can withstand a maximum tension of 43 N without breaking. A child ties a 0.37 kg stone to one end and, holding the other end, whirls the stone in a vertical circle of radius 0.91 m, slowly increasing the speed until the string breaks. Where is the stone on its path when the string breaks? at a random point, lowest point, or highest point on the path, or cannot be determined? What is the speed of the stone as the string breaks?

To determine where the stone is on its path when the string breaks, we need to consider the tension in the string at different points.

As the stone moves in a vertical circle, the tension in the string is highest at the lowest point and lowest at the highest point. This is because at the lowest point, the tension in the string needs to support the weight of the stone as well as provide the centripetal force required for circular motion. At the highest point, the tension only needs to provide the centripetal force.

Since the maximum tension the string can withstand is 43 N, the string will break at the point where the tension exceeds this value. Therefore, the string will break at the lowest point because that's where the tension is highest.

Now, let's find the speed of the stone when the string breaks. At the lowest point, the tension in the string can be calculated using the equation:

Tension = mg + (m * v^2) / r

where
Tension = tension in the string
m = mass of the stone (0.37 kg)
g = acceleration due to gravity (9.8 m/s^2)
v = velocity of the stone
r = radius of the circular path (0.91 m)

Since the tension at the lowest point is the maximum tension the string can withstand (43 N), we can rearrange the equation to solve for the velocity:

43 = (0.37 * 9.8) + (0.37 * v^2) / 0.91

Simplifying the equation:

4.132 = 0.37 + (0.37 * v^2) / 0.91

Next, rearrange the equation to solve for v:

(0.37 * v^2) / 0.91 = 4.132 - 0.37

0.37 * v^2 = (4.132 - 0.37) * 0.91

v^2 = [(4.132 - 0.37) * 0.91] / 0.37

Finally, take the square root of both sides to find the velocity:

v = sqrt{[(4.132 - 0.37) * 0.91] / 0.37}

By solving this equation, you can find the velocity v at the moment the string breaks.