Two discs of same moment of inertia rotating about their regular axis passing through centre and perpendicular to the plane of disc with angular velocities ω$$1$$ and ω$$2$$. They are brought into contact face to face coinciding the axis of rotation. The expression for loss of energy during this process is

Question

Two discs of same moment of inertia rotating about their regular axis passing through centre and perpendicular to the plane of disc with angular velocities ω$$1$$ and ω$$2$$. They are brought into contact face to face coinciding the axis of rotation. The expression for loss of energy during this process is

Infinity Learn · Accepted Answer

Initial angular momentum  Iω$$1+I$$ω$$2$$ Let ω be angular speed of the combined system. Final angular momentum $$2I$$ω∴ According to conservation of angular momentum Iω$$1+I$$ω$$2=2I$$ω   or   ω$$=$$ω$$1+$$ω$$22$$ initial rotational kinetic energy $$Ei=12I$$ω$$12+$$ω$$22Final$$ rotational kinetic energy $$Ef=12(2I)$$ω$$2=12(2I)$$ω$$1+$$ω$$222=14I$$ω$$1+$$ω$$22$$∴ Loss of energy Δ$$E=Ei-Ef=I2$$ω$$12+$$ω$$22-I4$$ω$$12+$$ω$$22+2$$ω$$1$$ω$$2=I2$$ω$$12+$$ω$$22-$$ω$$12+$$ω$$22+2$$ω$$1$$ω$$22=I22$$ω$$12+2$$ω$$22-$$ω$$12-$$ω$$22-2$$ω$$1$$ω$$22=I4$$ω$$12+$$ω$$22-2$$ω$$1$$ω$$2=I4$$ω$$1-$$ω$$22$$

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