Three particles, each of mass 4.0 kg , are fastened to each other and to a rotat

Question

Three particles, each of mass 4.0 kg, are fastened to each other and to a rotation axis by three massless strings, each 0.310 m long, as shown in the Figure. The combination rotates around the rotation axis with an angular velocity of 19.0 rad/s in such a way that the particles remain in a straight line. What is the rotational inertia of the system?

(I used I= 1/3(3L)^2+m(L)^2+m(2L)^2+m(3L)^2 but it was wrong)

What is the magnitude of the total angular momentum of the three masses?

If the outermost mass was to break free from the string, what would its linear momentum be?

Explanation / Answer

From the definition of the momentum of inertia, I = I1 + I2 +I3

and the moment of inertia of each particle, I = mR^2

I1 = 4(0.310)^2 = 0.3844 kg.m^2

I2 = 4(0.620)^2 = 1.5376 kg.m^2

I3 = 4(0.930)^2 = 3.4596 kg.m^2

therefore the total momentum of the system about the axies, I = I1 + I2 +I3

I = 5.3816 kg.m^2

the tansential velocity of the last particle V = rw = 0.930*19.0 = 17.67 m/s

the momentum of the last particle P = mv = 4*(17.67) = 70.68 kg.m/s.

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