two discs are located on the same axis the lower disc i1(8kgm^2is initialy rottating with angular velocity 4rad/sec and the upper one i2(4kg/m2 )is not rottating the upper disc then drops in the lower disc and stic togther and reach the same angular velocity find the common velocity
Let the final common angular velocity be ω. Since there is no external torque acting on the system, the angular momentum is conserved. Therefore, the initial angular momentum equals the final angular momentum.
Initial Angular Momentum (L_initial) = I1 * ω1 + I2 * ω2
where I1 = 8 kg m^2, ω1 = 4 rad/s, I2 = 4 kg m^2, and ω2 = 0 rad/s (since the upper disk is not rotating initially).
L_initial = (8 kg m^2)(4 rad/s) + (4 kg m^2)(0 rad/s) = 32 kg m^2/s.
Final Angular Momentum (L_final) = (I1 + I2) * ω
where I1 + I2 = 12 kg m^2 (since the disks stick together).
From the conservation of angular momentum, L_initial = L_final:
32 kg m^2/s = (12 kg m^2) * ω
Now we can solve for the final common angular velocity:
ω = (32 kg m^2/s) / (12 kg m^2) = 2.67 rad/s (approximately).
So, the common angular velocity is approximately 2.67 rad/s.