How to calculate tension in a cord when the cord has mass?

Question

I know how to deal with tension in a massless cord, but what if the cord has mass? Here is the specific question:

A block of mass M is attached to a cord of mass m and length , which is fixed at one end. The block moves in a horizontal circle on a frictionless table. If the period of the circular motion is P, find the tension in the cord as a function of radial position along the cord, 0 <= r <= L.

Apparently an integral needs to be set up, but I don't know how to approach it.

Sorry, the cord has a length of L.

Answer 1

You can do the following.
1. Angular velocity w is constant = 2pi/P
2. At the tip of the cord, the tension is equal to the centripetal force pulling on M, T(M)= MLw^2, this will stay constant

3. For the rope, let p= linear mass density = m/L
Consider a small mass dm at distance x from the opposite end of the cord, then integrate the force

F (r) = Integrate from r to L{ (L-x)w^2 dm} + T(M)
= Int from r toL { mw^2/L * (L-x)dx + TM
= mw^2/L * { L(L-r) - 1/2 * (L^2- r^2)} + T(M)
= mw^2/L * 1/2 * (L-r)^2 + T(M)

Test answer
a. r=L, F(r=L) = T(M)
b. r=0, F(r=0) = T(M) +1/2* mLw^2

The 1/2L is the center of mass of the cord so it is as if the cord of mass is located at 1/2L, together with the M at L.

Answer 2

I of the disc = Mp r²/2 angular acceleration of the disc ? = a /r the place a is the linear acceleration. Torque ? = I ? = Mp r a /2 = F*r the place F is the rigidity in the string M g = Mp a /2 a million*9.8 = 4 a /2 a = 4.9 m/s² ===============