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?

Split into 2 scenarios

1. Fc=T

so find mass using above formula

((mv^2)/r)=T
m=0.3kg

2. Find new tension with new values of velocity and radius and mass

Fc=t
((mv^2)/r)=T
T=780.3

To find the breaking strength of the string, we need to analyze the different forces acting on the rock at the point when the string breaks.

First, let's calculate the centripetal force acting on the rock when the string is 1.00 m long. The centripetal force is provided by the tension in the string and is given by the formula:

Fc = m * v^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 circular path

Since we are only interested in the breaking strength of the string, we can assume that the mass of the rock is negligible compared to the other forces acting on the rock.

Substituting the given values into the formula, we get:

Fc = (m * v^2) / r

Now let's calculate the centripetal force at 1.00 m length:

Fc = (m * (51.0 m/s)^2) / 1.00 m

Next, we need to calculate the tension in the string when the string is 1.00 m long and the speed of the rock is 51.0 m/s. We can use the tension formula:

T = (m * v^2) / r + mg

Where:
T is the tension in the string
m is the mass of the rock
v is the speed of the rock
r is the radius of the circular path
g is the acceleration due to gravity (9.8 m/s^2)

Since the mass of the rock is negligible, we can ignore the mg term in our calculations.

Substituting the given values into the formula, we get:

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

Now we know that the string breaks when the tension exceeds its breaking strength. Therefore, the breaking strength of the string is equal to the tension in the string, which we just calculated.

To find the breaking strength of the string, we need to determine the maximum tension it experienced before breaking.

We can start by calculating the centripetal force acting on the rock at the maximum speed.
The centripetal force, Fc, is given by the formula:

Fc = (m * v^2) / r

Where:
m = mass of the rock
v = velocity of the rock
r = radius of the circular path

To determine the mass of the rock, we need to use the formula:

Fc = m * g

Where:
g = acceleration due to gravity

Solving for m:

m = Fc / g

Now, let's substitute the given values:
Fc = 50.0 N (tension under which the rock starts whirling)
r = 2.50 m
v = 20.4 m/s
g = 9.8 m/s^2

Upon substituting the values, we obtain:

m = (50.0 N) / (9.8 m/s^2)

m ≈ 5.10 kg

Now let's calculate the centripetal force at the point when the string broke, which is given by:

Fc' = (m * v'^2) / r'

Where:
v' = final velocity of the rock
r' = length of the string at the time of breaking

Given values:
v' = 51.0 m/s
r' = 1.00 m

Substituting the values:

Fc' = (5.10 kg * (51.0 m/s)^2) / 1.00 m

Fc' ≈ 13050 N

Therefore, the breaking strength of the string is approximately 13050 N.