kg and radius 2 m is rotating about an axis passing through its centre and perpendicular to its plane. The work done to increase the angular speed of the disc from 90rev∕min to 120rev∕min is
Step 1: Find the moment of inertia of the disc. The moment of inertia ( I ) for a disc about its center is I=‌
1
2
MR2 Here, the mass M=‌
10
Ï€2
kg, and the radius R=2m. So, I=‌
1
2
×‌
10
Ï€2
×(2)2=‌
1
2
×‌
10
Ï€2
×4=‌
20
Ï€2
kgm2 Step 2: Change revolutions per minute (rev/min) to angular speed (rad/s). The first speed is 90rev∕min. To convert to radians per second: w1=90×‌
2Ï€
60
=3πrad∕s The second speed is 120rev∕min:w2=120×‌
2Ï€
60
=4πrad∕s Step 3: Use the rotational kinetic energy formula to find work done. The work done to increase speed equals the change in rotational kinetic energy: Work Done =∆KE=‌
1
2
I(w22−w12) Plug in the values: Work =‌
1
2
×‌
20
Ï€2
[(4π)2−(3π)2] Calculate: ‌(4π)2=16π2 ‌(3π)2=9π2 So, 16π2−9π2=7π2 Therefore, Work =‌