A string under a tension of 43 N is used to whirl a rock in a horizontal circle of radius 2.1 m at a speed of 15.89 m/s. The string is pulled in, and the speed of the rock increases. When the string is 0.948 m long and the speed of the rock is 59 m/s, the string breaks.
What is the breaking strength of the string? Answer in units of N.
f = m v^2 / r
f = 43 N * (59/15.89)^2 / (.948/2.1)
To find the breaking strength of the string, we can use the concept of centripetal force. The tension in the string provides the necessary centripetal force to keep the rock moving in a circle.
We can start by calculating the centripetal force at the first instant when the speed of the rock is 15.89 m/s. The centripetal force is given by the formula:
F = (m * v^2) / r
Where:
F is the centripetal force
m is the mass of the rock
v is the velocity of the rock
r is the radius of the circle
From the given information, we know the radius (r = 2.1 m) and the speed (v = 15.89 m/s). However, we don't have the mass of the rock. Therefore, we need to find it.
To calculate the mass of the rock, we can use the fact that weight (W) is equal to the force of gravity on the rock. Weight is given by the formula:
W = m * g
Where:
m is the mass
g is the acceleration due to gravity (approximately 9.8 m/s^2)
Since we know the weight of the rock (W = 43 N), we can calculate its mass by rearranging the formula:
m = W / g
Plugging in the values, we get:
m = 43 N / 9.8 m/s^2 = 4.39 kg
Now, we can calculate the centripetal force using the mass we just found:
F = (m * v^2) / r
F = (4.39 kg * (15.89 m/s)^2) / 2.1 m
F ≈ 425.39 N
Therefore, at the first instant, the tension in the string is approximately 425.39 N.
Now, let's move on to the second situation when the string is 0.948 m long and the speed of the rock is 59 m/s.
In this case, the tension in the string is equal to the breaking strength of the string. We can calculate it using the same centripetal force formula:
F = (m * v^2) / r
Now, we have the velocity (v = 59 m/s) and the radius (r = 0.948 m). We will use the previously calculated mass (m = 4.39 kg):
F = (4.39 kg * (59 m/s)^2) / 0.948 m
F ≈ 16359.231 N
Therefore, the breaking strength of the string is approximately 16359.231 N.