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?
I still can't get this one. First I tried adding the gravitational force of the earth, 9.8*.37, with the Tension, 43N, and then dividing by the radius and taking the square root to find the velocity but the answer was incorrect. Any suggestions???
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.
First, let's find the tension when the stone is at the lowest point of its path. At the lowest point, the tension in the string is equal to the sum of the weight of the stone and the centripetal force required to keep it moving in a circle.
The weight of the stone can be calculated as:
Weight = mass * acceleration due to gravity = 0.37 kg * 9.8 m/s^2 = 3.626 N.
The centripetal force can be calculated as:
Centripetal Force = mass * (velocity^2 / radius of the circle)
At the lowest point, the velocity of the stone is the maximum, and the tension in the string is at its highest. So, to find out where the string will break, we have to compare the maximum tension of 43 N with the calculated tension at the lowest point.
Now, let's calculate the tension at the lowest point of the path:
Tension = Weight + Centripetal Force
Tension = 3.626 N + (0.37 kg * (velocity^2 / 0.91 m))
To find the speed of the stone as the string breaks, we need to solve for velocity. Set the tension equal to the maximum tension of 43 N and solve for velocity:
43 N = 3.626 N + (0.37 kg * (velocity^2 / 0.91 m))
Rearrange the equation to solve for velocity:
0.37 kg * (velocity^2 / 0.91 m) = 43 N - 3.626 N
0.37 kg * (velocity^2 / 0.91 m) = 39.374 N
velocity^2 / 0.91 m = (39.374 N) / (0.37 kg)
velocity^2 = (39.374 N * 0.91 m) / 0.37 kg
velocity^2 = 96.778 Nm / 0.37 kg
velocity^2 = 261.561 Nm/kg
Taking the square root of both sides, we find:
velocity ≈ √(261.561 Nm/kg) ≈ 16.181 m/s (rounded to three decimal places)
After calculating the velocity, we need to check where the stone is on its path when the string breaks. Since we found that the maximum tension occurs at the lowest point, the stone will break from the string at the highest point of its path.
Therefore, the stone is at the highest point on its path when the string breaks, and the speed of the stone as the string breaks is approximately 16.181 m/s.