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Answer :
The final angular velocity of the disks would be 24.3 rad/s.
To resolve this issue, can employ energy conservation. The system's potential energy is transformed into kinetic energy when the discs are released from their resting state, which causes them to begin rotating. To determine the final angular velocity of the discs, we can set the initial potential energy equal to the final kinetic energy.
The potential energy of the system is given by:
U = mgh
where m is the disk's mass, g is its gravitational acceleration, and h is its height above a reference point. In this instance, we can consider the reference level to be the height of the disk's centre of mass, which is located r/2 away from the disk's centre. As a result, the disk's height above the reference level is:
h = r/2
The total potential energy of the system is then:
U = 2mg*(r/2) = mgr
where we have multiplied by 2 because there are two disks.
The kinetic energy of a rotating object is given by:
K = (1/2)Iω²
where I is the moment of inertia of the object and ω is the angular velocity. For a disk rotating about its center, the moment of inertia is:
I = (1/2)mr²
Thus, the total kinetic energy of the system is:
K = (1/2)2(1/2)mr²ω² = (1/2)mr²ω²
where we have multiplied by 2 because there are two disks, and by (1/2) because the cord is wrapped around the disk halfway.
By conservation of energy, the initial potential energy must equal the final kinetic energy:
U = K
mgr = (1/2)mr²ω²
Solving for ω, we find:
ω = √(2g/r)
Substituting the given values, we have:
ω = √(2*9.81/0.084) = 24.3 rad/s
Therefore, the final angular velocity of the disks is 24.3 rad/s.
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