The loss of kinetic energy in the above process is [2007]

Question

Two discs A and B are mounted coaxially on a vertical axle. The discs have moments of inertia $I$ and $2 I$ respectively about the common axis. Disc A is imparted an initial angular velocity $2 ω$ using the entire potential energy of a spring compressed by a distance $x_{1}$ . Disc B is imparted an angular velocity $ω$ by a spring having the same spring constant and compressed by a distance $x_{2}$ . Both the discs rotate in the clockwise direction. [2007]

Q. The loss of kinetic energy in the above process is

Smart Achievers · Accepted Answer

(2)

$Loss in kinetic energy = {(K . E .)}_{initial} - {(K . E .)}_{final}$

$= [\frac{1}{2} I {(2 ω)}^{2} + \frac{1}{2} (2 I) ω^{2}] - [\frac{1}{2} (I + 2 I) {(\frac{4}{3} ω)}^{2}]$

$= 3 I ω^{2} - \frac{8}{3} I ω^{2} = \frac{I ω^{2}}{3}$

Solution

1 $\frac{I ω^{2}}{2}$

2 $\frac{I ω^{2}}{3}$

3 $\frac{I ω^{2}}{4}$

4 $\frac{I ω^{2}}{6}$

Ans.
(2)

$Loss in kinetic energy = {(K . E .)}_{initial} - {(K . E .)}_{final}$

$= [\frac{1}{2} I {(2 ω)}^{2} + \frac{1}{2} (2 I) ω^{2}] - [\frac{1}{2} (I + 2 I) {(\frac{4}{3} ω)}^{2}]$

$= 3 I ω^{2} - \frac{8}{3} I ω^{2} = \frac{I ω^{2}}{3}$

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Solution

1 Iω22

2 Iω23

3 Iω24

4 Iω26