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In the above question what would be the loss in K.E. of system in the process? |
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Answer» `(I_(1)I_(2)(omega_(1)-omega_(2))^(2))/(2(I_(1)+I_(2)))` `K_(i)=(1)/(2)I_(1)omega_(1)^(2)+(1)/(2)I_(2)omega_(2)^(2)` `=(1)/(2)[I_(1)omega_(1)^(2)+I_(2)omega_(2)^(2)]` Also `K_(f)=(1)/(2)(I_(1)+I_(2))((I_(1)omega_(1)+I_(2)omega_(2))^(2))/((I_(1)+I_(2))^(2))` `=(1)/(2)((I_(1)omega_(1)+I_(2)omega_(2))^(2))/((I_(1)+I_(2)))` and `K_(f)-K_(i)=(1)/(2)((I_(1)^(2)omega_(1)^(2)+I_(2)^(2)omega_(2)^(2)+2I_(1)I_(2)omega_(1)omega_(2)))/(I_(1)+I_(2))-[(1)/(2)I_(1)omega_(1)^(2)+(1)/(2)I_(2)omega_(2)^(2)]` = `(1)/(2){(I_(1)^(2)omega_(1)^(2)+I_(2)^(2)omega_(2)^(2)-I_(1)^(2)omega_(1)^(2)-I_(2)^(2)omega_(2)^(2)-I_(1)I_(2)(omega_(1)^(2)+omega_(2)^(2))+2I_(1)I_(2)omega_(1)omega_(2))/(I_(1)+I_(2))}` `=-(1)/(2){(I_(1)I_(2)(omega_(1)^(2)+omega_(2)^(2)-2omega_(1)-omega_(2)))/(I_(1)+I_(2))}` `=-(1)/(2){(I_(1)I_(2))/(I_(1)+I_(2))(omega_(1)+omega_(2))^(2)}` Now `(I_(1)I_(2))/(I_(1)+I_(2))` cannot be - ve. Also `(omega_(1)-omega_(2))^(2)` cannot be - ve being SQUARE. Thus over all difference `(K_(f)-K_(i))` is - ve. There is a LOSS in K.E which is due to the work done against the friction between two DISC on CONTACT. |
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