A string under a tension of 43.0 N is used to whirl a rock in a horizontal circle of radius 2.25 m at a speed of 20.0 m/s on a frictionless surface as shown in the figure below. As the string is pulled in, the speed of the rock increases. When the string on the table is 1.00 m long and the speed of the rock is 49.5 m/s, the string breaks. What is the breaking strength, in newtons, of the string?
T = mv^2/r
Use the initial info to solve for m
m = Tr/v^2
Then plug back into the original eq to find T
To find the breaking strength of the string, we need to determine the maximum tension force the string can withstand.
Let's analyze the forces acting on the rock when the string breaks. At that moment, the tension force in the string is equal to its breaking strength, and the centripetal force acting on the rock is providing the acceleration needed to keep the rock moving in a circle.
The centripetal force is given by the equation:
Fc = (mv^2) / r
Where:
- Fc is the centripetal force
- m is the mass of the rock
- v is the speed of the rock
- r is the radius of the circle
We need to find the mass of the rock first. We can use the formula:
F = ma
Where:
- F is the force acting on the rock (in this case, the tension force)
- a is the acceleration
Since the rock is moving in a circle at a constant speed, the acceleration is given by:
a = (v^2) / r
Substituting this acceleration back into the force equation gives us:
F = m(v^2) / r
Now we can solve for the mass by rearranging the equation:
m = F * r / (v^2)
We substitute this mass value back into the centripetal force equation:
Fc = [(F * r / (v^2)) * v^2] / r
Fc = F
By substituting the values into the centripetal force equation, we find that the breaking strength of the string is equal to the centripetal force acting on the rock when the string breaks. Therefore, the breaking strength of the string is equal to the centripetal force required to keep the rock moving in a circle of radius 1.00 m at a speed of 49.5 m/s.
Using the centripetal force equation:
Fc = (m * v^2) / r
Substituting the known values:
Fc = [(F * r / (v^2)) * v^2] / r
Fc = (43.0 N * 2.25 m) / (20.0 m/s)^2
Calculating this expression gives us the answer.