A table with smooth horizontal surface is placed in a cabin which moves in a circle of large radius R (figure 7-35) A smooth pulley of small radius is fastened to the table. Two masses m and 2m placed on the table are connected through a string going over the pulley. Initially the masses are held by a person with the strings along the ouward radius and then the system is released from rest Find the magnitude of the initial acceleration of the masses as seen from the cabin and the tension in the string
Since, we need to find the Acceleration and Tension with respect to the Cabin then considered that we are sitting in the Non-Inertial Frame of the Reference.
Now, If we were in the cabin, then we will feel the centrifugal force outside.
∴ There must be the Centripetal Force on the masses attached with the Pulleys.
Let the Tension in the Masses be 'T' and acceleration be 'a'.
Now, Let us assume that the mass m is going upwards and mass 2m is going downwards. Also, Centrifugal force equal to mω²R will be acting outwards on both masses.
∴ In mass $$m, (m_1)$$
$$T - mw^2R = ma$$
⇒ $$T = mw^2R + ma$$.
Similarly, In mass $$2m$$,
$$2mw^2r - T = 2ma$$
⇒ $$T = 2mw^2R - 2ma$$
On solving, it we will get,
$$a = w^2R/3$$ and
$$T = \dfrac{4}{3} \times mw^2R$$
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