A straight, thin and homogeneous bar of length l and mass m is in a quiet positi

Question

A straight, thin and homogeneous bar of length l and mass m is in a quiet position on a horizontal plane. A hammer blow exerts an impulse J given at one end of the bar and orthogonally to it. Calculate the kinetic energy of the bar after the application of the impulse

b) Una sbarra rettilinea, sottile ed omogenea di lunghezza l e massa m si trova in quiete su un piano orizzontale. Un colpo di martello esercita un impulso J dato ad un estremo della sbarra e ortogonalmente ad essa. Calcolare l'energia cinetica della sbarra dopo l'applicazione dell'impulso. [Siano noti m, I, il momento di inerzia dell'asta Icm-1/12ml e J.)

Explanation / Answer

Using impulse - momentum relation

J = Iw

J = Impulse; w = angular velocity gained
Here I = mL^2 /12 + m x (L/2)^2 = mL^2 [1/12 + 1/4] = mL^2/3 .... because it is hammered at the end

But KE = 1/2 I w^2

= 1/2 I x (J/I)^2 = 1/2 J^2/I

= 1/2 J2 / mL2/3

= 3/2 J2/mL2

Hope it helps

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