A string under a tension of 50.0 N is used to whirl a

rock in a horizontal circle of radius 2.50 m at a speed of
20.4 m/s. The string is pulled in and the speed of the
rock increases. When the string is 1.00 m long and the
speed of the rock is 51.0 m/s, the string breaks. What is
the breaking strength (in newtons) of the string?

To find the breaking strength of the string, we need to calculate the centripetal force acting on the rock at the moment the string breaks.

We can use the centripetal force formula:

F = (m * v^2) / r

where:
F is the centripetal force,
m is the mass of the rock,
v is the speed of the rock,
and r is the radius of the circle.

First, let's calculate the mass of the rock. We are given the tension in the string, which is equal to the centripetal force:

T = F

Therefore, we can substitute T for F:

T = (m * v^2) / r

Rearranging the equation, we can solve for m:

m = (T * r) / v^2

Now we can calculate the mass:

m = (50.0 N * 2.50 m) / (20.4 m/s)^2

m = 2.44 kg (rounded to 2 decimal places)

Next, we need to calculate the new tension in the string before it breaks. Using the same formula:

T = (m * v^2) / r

Substituting the given values:

T = (2.44 kg * (51.0 m/s)^2) / 1.00 m

T = 6291.24 N (rounded to 2 decimal places)

Therefore, the breaking strength of the string is approximately 6291.24 newtons.