How Do You Calculate Tension in a Two-Rock System in Circular Motion?

[quickclick330]

Its just one of those days where my brain needs a bit of a jump start, if anyone could help me it would be greatly appreciated!Thanks!

Problem:

A rock of mass m1 = 0.4 kg is tied to another rock with a mass m2 = 0.58 kg with a string of length L1 = 0.14 m. The rock m2 is tied to another string of length L2 = 0.19 m, and the pair of rocks is swung around in uniform circular motion, making 2 complete revolutions in one second. In this problem, you should neglect gravity and assume the motion is in the horizontal plane.


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a) What is T2, the tension in the string that is connected to the rock with mass m2?


My attempt
I tried taking v = (2*pi*0.33)/.5 seconds = 4.14 m/s
then: T = ((0.58 kg + 0.4 kg)*(4.14 m/s)^2)/0.19 = 88.40 N
but it was wrong...help! thank you!

 

[berkeman]

Think of it as two separate problems, and then add the tensions. What is the tension in a string L1+L2 long that has m1 at the end, with the angular frequency 2x2PI radians per second? And what is the tension in a string L2 long with the mass m2 at the end, with the same angular frequency? (BTW, I'm assuming that one end of L2 is what is being held to swing the system around.)

What is the equation that gives the force (tension) required for uniform circular motion of a mass, in terms of the mass m, the angular frequency omega, and the radius R?

 

[m2003]

It is commendable that you are attempting to solve this problem and actively seeking help when needed. Let's take a closer look at the situation to better understand the problem and find the correct solution.

Firstly, we need to consider the forces acting on the rocks in this scenario. Since gravity is neglected, the only force acting on the rocks is the tension in the strings. In circular motion, the centripetal force (Fc) is equal to the tension in the string (T) and can be calculated using the formula Fc = mv^2/r, where m is the mass of the object, v is the velocity, and r is the radius of the circular motion.

In this case, the radius of the circular motion is the length of the string, which is given by L1 + L2 = 0.14 m + 0.19 m = 0.33 m. Using this value, we can calculate the centripetal force as Fc = ((0.58 kg + 0.4 kg)* (4.14 m/s)^2)/0.33 m = 88.40 N. This is the correct value for the tension in the string connected to the rock with mass m2.

It is important to note that the velocity used in this calculation is the tangential velocity, which is the speed of the object as it moves along the circular path. This is different from the linear velocity, which is the distance traveled per unit time. In this case, the linear velocity would be 2πr/1 s = 2π*0.33 m/1 s = 2.07 m/s. This is the value that was used in your attempt, which resulted in an incorrect answer.

In summary, to find the tension in the string connected to the rock with mass m2, we need to use the formula Fc = mv^2/r and calculate the centripetal force using the tangential velocity and the radius of the circular motion. I hope this helps to jump start your brain and solve the problem. Keep up the good work!