A string under a tension of 48 N is used to whirl a rock in a horizontal circle of radius 2.4 m at a speed of 18.48 m/s. The string is pulled in, and the speed of the rock increases.
When the string is 1.076 m long and the speed of the rock is 49.4 m/s, the string breaks. What is the breaking strength of the string?
Answer in units of N.
The string tension is proportional to V^2/R, the centripetal acceleration.
[V^2/R]initial = 142.3 m/s^2
[V^2/R]final = 2268 m/s^2
The breaking strength is therefore
48*(2268/142.3) = ___ N
765.03
To find the breaking strength of the string, we need to calculate the maximum tension force that the string can handle.
The tension force in the string is given by the centripetal force required to keep the rock moving in a circular path. The centripetal force is given by the equation:
F = (m * v^2) / r
Where:
F is the centripetal force (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 know the tension force acting on the string is 48 N. At 18.48 m/s, the rock is moving in a circular path with a radius of 2.4 m. The centripetal force is equal to the tension force:
48 N = (m * 18.48^2 m^2/s^2) / 2.4 m
To find the mass, rearrange the equation:
m = (48 N * 2.4 m) / 18.48^2 m^2/s^2
m ≈ 1.62 kg
Now that we know the mass of the rock, we can calculate the centripetal force when the string breaks. At this point, the string is 1.076 m long and the rock is moving at a speed of 49.4 m/s:
F = (1.62 kg * 49.4^2 m^2/s^2) / 1.076 m
F ≈ 361.38 N
Therefore, the breaking strength of the string is approximately 361.38 N.