a horizontal disc rotating freely about a vertical axis make 100 revolution per minute. a small piece of wax of Mass 10g falls vertically UN to the disc and to it a distance of 9cm form the axis. if the number of revolution per minute is their by returned to go ,calculated the number moment of inertial of the disc
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To calculate the moment of inertia of the disc, we need to use the conservation of angular momentum.
Angular momentum (L) is defined as the product of moment of inertia (I) and angular velocity (ω). Therefore, when the piece of wax falls onto the disc, the change in angular momentum is equal to the angular momentum acquired by the wax.
The angular momentum before the wax falls onto the disc is given by:
L1 = I1 * ω1
where I1 is the initial moment of inertia of the disc and ω1 is the initial angular velocity (100 revolutions per minute).
When the wax falls onto the disc, its moment of inertia changes to I2 and the angular velocity changes to ω2 (unknown value). The angular momentum after the wax falls onto the disc is given by:
L2 = (I1 + I2) * ω2
Since there is no external torque acting on the system, the conservation of angular momentum states that L1 = L2. Therefore:
I1 * ω1 = (I1 + I2) * ω2
We are given that the wax falls to a distance of 9 cm from the axis, so its moment of inertia (Iwax) can be calculated using the parallel axis theorem:
Iwax = mwax * r^2
where mwax is the mass of the wax (10g or 0.01 kg) and r is the distance of the wax from the axis (9 cm or 0.09 m).
Substituting the values into the equation, we have:
I1 * 100 revolutions per minute = (I1 + Iwax) * ω2
I1 * 2π * 100 radians per minute = (I1 + Iwax) * 2π * ω2
Simplifying the equation, we have:
I1 * 100 = (I1 + Iwax) * ω2
Now we have the equation to calculate the moment of inertia of the disc (I1):
I1 = (I1 + Iwax) * ω2 / 100
Rearranging the equation, we can solve for I1:
I1 - (I1 + Iwax) * ω2 / 100 = 0
I1 - I1 * ω2 / 100 - Iwax * ω2 / 100 = 0
I1 * (1 - ω2 / 100) - Iwax * ω2 / 100 = 0
I1 * (1 - ω2 / 100) = Iwax * ω2 / 100
I1 = (Iwax * ω2 / 100) / (1 - ω2 / 100)
Substituting the values of Iwax and ω2 (to be determined), we can calculate I1.