A stone of mass 3.2 kg is tied to a string of length 0.9 m, and is swung in a horizontal circle with speed v. The string has a breaking point force of 10.1 N. What is the largest value that v can have without breaking the string?
To find the largest value of speed (v) without breaking the string, we need to consider the tension force acting on the string when the stone is in motion.
When an object moves in a circular path, it experiences a centripetal force that keeps it moving in a circle. In this case, the centripetal force is provided by the tension force in the string. The centripetal force (Fcp) can be calculated using the following formula:
Fcp = (m * v^2) / r
Where:
m = mass of the stone = 3.2 kg
v = speed of the stone
r = radius of the circular path = length of the string = 0.9 m
We know that the breaking point force of the string is 10.1 N. So, the tension force (T) should not exceed this value.
To find the largest value of v without breaking the string, we need to find the maximum tension force (Tmax) that the string can withstand. Rearranging the formula for centripetal force gives us:
T = (m * v^2) / r
By substituting the given values, we get:
10.1 N = (3.2 kg * v^2) / 0.9 m
Now, let's solve the equation to find the maximum value of v:
10.1 N * 0.9 m = 3.2 kg * v^2
v^2 = (10.1 N * 0.9 m) / 3.2 kg
v^2 = 2.84 m^2/s^2
v ≈ √(2.84 m^2/s^2)
v ≈ 1.69 m/s
Therefore, the largest value that v can have without breaking the string is approximately 1.69 m/s.