A string under a tension of 43 N is used to whirl a rock in a horizontal circle of radius 3.3 m at a speed of 26.46 m/s. The string is pulled in, and the speed of the rock increases. When the string is 0.635 m long and the speed of the rock is 80.5 m/s, the string breaks. What is the breaking strength of the string? Answer in units of N.
Let's make the following denominations:
F1 = 43 N R1 = 3.3 m V1 = 26.46 m/s R2 = 0.635 m V2 = 80.5 m/s M - the mass of a rock
We need to find the mass of a rock at first. Using the first case we have that the centripetal acceleration of a rock is Ac1 = V1²/R1 = 26.46²/3.3 ≈ 212.1611 m/s². Now let's use the second Newton's law: F1 = M*Ac1 ==> M = F1/Ac1 = 43/212.1611 = 0.2027 kg.
Now let's find the breaking strength of the string. Using the second case we have that the centripetal acceleration of a rock is Ac2 = V2²/R2 = 80.5²/0.635 ≈ 10205.1181 m/s². So, the breaking strength is F2 = M*Ac2 = 0.2027*10205.1181 = 2068.5774 N.