Question

Derive an algebraic expression for the tension in the segment of the cord between the post and block A (assumed massless). Express your answer in terms of the variables mA, mB, rA, rB, f, and appropriate constants.

Answer 1

Answer: T = 4·$\pi ^{2}$·f (mA · rA + mB · rB)

Explanation: According to Newton's First Law of motion, an object stays in the same state of motion unless a resultant force acts on it. If the resultant force is zero, the object stays stationery or in movement with the same velocity. In the system described, the two blocks are moving in the same frequency (f). For the block A, the forces acting on it are Centripetal Force on block A (FcA), Tension of the cord and Centripetal Force of block B (FcB). This last one counts because the two blocks are connected to each other with the same cord. So, the Resultant Force will be

FcA+ FcB + T = 0

Now, Centripetal Forces always points to the center of the circular movement.

Assuming that pointing towards the center is positive, the expression will be: FcA+ FcB - T = 0, in which tension (T) acts opposite to the others.

To calculate Fc, the expression is

Fc = m · ac

The term ac is the centripetal acceleration and is expressed as

ac = $\frac{v^{2} }{R}$ and can also be written as ac = ω²·R.

As ω is the angular velocity and can be written as ω = 2·$\pi$·f, we have

ac = (2·$\pi$·f)²·R

So, FcA+ FcB - T = 0

Calculating and substituting:

- T = - FcA - FcB

T = FcA + FcB

T = mA·(2·$\pi$·f)²·rA· + mB·(2·$\pi$·f)²·rB

T = 4·$\pi ^{2}$·f (mA · rA + mB · rB)

Answer 2

T=m1a+m2a 

a=v^2/r v=r*w w=2πF--->a=r(2f)^2 <r1 for m1 and r2 for m2> 

T=(m1r1+m2r2)(2πf)^2 <i factored out (2πf)^2