A string under a tension of 68 N is used to whirl a rock in a horizontal circle of radius 3.7 m at a speed of 16.53 m/s. The string is pulled in, and the speed of the rock increases. When the string is 0.896 m long and the speed of the rock is 71.5 m/s, the string breaks.

What is the breaking strength of the string? Answer in units of N

To find the breaking strength of the string, we need to calculate the tension in the string when it breaks. We can do this by applying the centripetal force formula.

The centripetal force acting on an object moving in a circular path is given by the formula:

F_c = (m v^2) / r

Where:
- F_c is the centripetal force
- m is the mass of the object
- v is the velocity of the object
- r is the radius of the circular path

In this case, we are given the tension in the string and the radius of the circular path. We can use this information to find the centripetal force acting on the rock at its maximum speed (71.5 m/s).

Since the string breaks when it is 0.896 m long, we can use this length to calculate the radius of the circular path when the string breaks.

We can use the formula for the length of a circular arc:

s = rθ

Where:
- s is the length of the arc
- r is the radius of the circular path
- θ is the angle subtended by the arc (in radians)

Given that the string length is 0.896 m, we can calculate θ:

θ = s / r
θ = 0.896 m / 3.7 m

Now we know the angle subtended by the arc (θ), we can use this information to calculate the radius (r) of the circular path when the string breaks. We can rearrange the equation θ = s / r to solve for r:

r = s / θ

Now that we have the radius (r = 3.7 m) and the velocity (v = 71.5 m/s) at which the string breaks, we can calculate the centripetal force (F_c) using the formula:

F_c = (m v^2) / r

Substituting the values, we have:

F_c = (m (71.5 m/s)^2) / 3.7 m

Since the mass (m) of the rock is not given, we can cancel it out by dividing both sides of the equation by m:

F_c / m = (71.5 m/s)^2 / 3.7 m

Now, we can solve for F_c / m:

F_c / m = (71.5 m/s)^2 / 3.7 m

Finally, since the breaking strength of the string is equal to the tension (F_c), we can find the breaking strength in units of N:

Breaking strength of the string = F_c = (71.5 m/s)^2 / 3.7 m

Calculating this equation will yield the breaking strength of the string in units of N.

To find the breaking strength of the string, we need to determine the tension in the string when it breaks. We can use the centripetal force equation to find the tension:

T = (m * v²) / r

Where:
T is the tension
m is the mass of the rock
v is the velocity of the rock
r is the radius of the circular path

First, let's find the mass of the rock. We can assume the mass remains constant throughout the problem.

Given:
Tension (T) = 68 N
Initial velocity (v₁) = 16.53 m/s
Initial radius (r₁) = 3.7 m

Using the centripetal force equation, we can rearrange it to solve for the mass:

m = (T * r₁) / v₁²

Substituting the given values:

m = (68 N * 3.7 m) / (16.53 m/s)²

m = (68 N * 3.7 m) / (272.9509 m²/s²)

m ≈ 0.69615 kg

Now that we have the mass of the rock, we can use the same centripetal force equation to find the tension just before the string breaks.

Final velocity (v₂) = 71.5 m/s
Final radius (r₂) = 0.896 m

T = (m * v₂²) / r₂

Substituting the values:

T = (0.69615 kg * (71.5 m/s)²) / 0.896 m

T = (0.69615 kg * 5112.25 m²/s²) / 0.896 m

T ≈ 3993.63 N

Therefore, the breaking strength of the string is approximately 3993.63 N.