A string under a tension of 42.6 N is used to whirl a rock in a horizontal circle of radius 2.47 m at a speed of 19.7 m/s. The string is pulled in and the speed of the rock increases. When the string is 1.13 m long and the speed of the rock is 46.9 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 point of breaking.
1. First, let's calculate the centripetal force when the rock is moving at a speed of 19.7 m/s with a string length of 2.47 m.
The centripetal force is given by the equation:
F = (m * v^2) / r
where F is the centripetal force, m is the mass of the rock, v is the velocity, and r is the radius.
2. We don't have the mass of the rock, but we can cancel it out by dividing both sides of the equation by m:
F / m = (v^2) / r
3. The left side of the equation represents the acceleration, so we can rewrite it as:
a = (v^2) / r
4. Now, we can substitute the given values into the equation:
a = (19.7^2) / 2.47
a = 155.609 N
5. Next, let's calculate the centripetal force when the rock is moving at a speed of 46.9 m/s with a string length of 1.13 m.
Using the same formula:
a = (v^2) / r
a = (46.9^2) / 1.13
a = 1936.052 N
6. The breaking strength of the string will be equal to the centripetal force at the point of breaking, which is 1936.052 N. Therefore, the breaking strength of the string is 1936.052 N.
To find the breaking strength of the string, we need to analyze the forces acting on the rock when the string breaks.
When the rock is moving in a circular path, there are two main forces acting on it: tension in the string and the centrifugal force.
The tension in the string provides the centripetal force required to keep the rock moving in a circle. The centripetal force is given by the equation:
Fc = (mass × velocity^2) / radius
where Fc is the centripetal force, mass is the mass of the rock, velocity is the speed of the rock, and radius is the radius of the circular path.
Hence, we can write:
Fc = (m × v^2) / r
We know the values for the radius and velocity when the string breaks (r = 1.13 m and v = 46.9 m/s), but we don't know the mass of the rock. To eliminate the mass from the equation, we can use the tension in the string.
The tension in the string is responsible for providing the centripetal force. Therefore, we can equate the tension with the centripetal force:
T = Fc = (m × v^2) / r
The tension in the string just before the string breaks is equal to the breaking strength of the string. So, we can calculate the breaking strength (Tb) as:
Tb = (m × v^2) / r
Now we need to find the mass of the rock. We can use the first set of given information to calculate the mass.
The tension in the string at the first instance is given as 42.6 N. So, we can use the equation:
T = (m × v^2) / r
Rearranging the equation, we get:
m = (T × r) / v^2
Substituting the values:
m = (42.6 N × 2.47 m) / (19.7 m/s)^2
Now we have the mass of the rock. We can substitute this value back into the formula to calculate the breaking strength:
Tb = (m × v^2) / r
Finally, substitute the value for mass (m) and the given values for speed (v) and radius (r) into the formula to calculate the breaking strength (Tb) in newtons.