A stone of mass 0.31 kg is tied to a string of length 0.65 m and is swung in a horizontal circle with speed v. The string has a breaking-point force of 70 N. What is the largest value v can have without the string breaking?
tension in the string = m*V^2/r
max tension is 70N
so Vmax = sqrt(70*r/m)
To determine the largest value of v without the string breaking, we need to consider the tension force in the string.
In this scenario, the tension force in the string provides the centripetal force required to keep the stone moving in a circular path. If the tension force exceeds the breaking-point force of the string, it will break.
The centripetal force can be calculated using the equation:
F_c = m * v^2 / r
Where:
F_c = Centripetal force
m = Mass of the stone (0.31 kg)
v = Speed of the stone
r = Length of the string (0.65 m)
We can rearrange the equation to solve for v:
v = sqrt(F_c * r / m)
Now, we know that the largest value of v without the string breaking is when the tension force is equal to the breaking-point force of the string.
Therefore, we can set F_c equal to the breaking-point force:
F_c = Breaking-point force = 70 N
Plugging in the given values:
v = sqrt(70 * 0.65 / 0.31)
Simplifying the equation:
v = sqrt(146.45 / 0.31)
v = sqrt(471.774)
Calculating the square root:
v ≈ 21.737 m/s
So, the largest value v can have without the string breaking is approximately 21.737 m/s.