A hoop of radius 2 m weighs 100 kg. It rolls along a horizontal floor so that its centre of mass has a speed of 20 cm/s. How much work has to be done to stop it?
Answers
Radius of the hoop, r = 2 m
Mass of the hoop, m = 100 kg
Velocity of the hoop, v = 20 cm/s = 0.2 m/s
Total energy of the hoop = Translational K.E. + Rotational K.E.
ET = (1/2)mv2 + (1/2) I ω2
Moment of inertia of the hoop about its centre, I = mr2
ET = (1/2)mv2 + (1/2) (mr2)ω2
But we have the relation, v = rω
∴ ET = (1/2)mv2 + (1/2)mr2ω2
= (1/2)mv2 + (1/2)mv2 = mv2
The work required to be done for stopping the hoop is equal to the total energy of the hoop.
∴ Required work to be done, W = mv2 = 100 × (0.2)2 = 4 J.
Given ,
Mass of hoop = 100 Kg
Radius = 2m
Speed of centre of mass = 20cm/s = 0.2 m/s
Work done by hoop to stop it = total kinetic energy of the hoop
= (1/2) mv² + (1/2)Iw²
Moment of inertia of hoop = mr² and w = v/r
Then,
Work done =(1/2)mv² + (1/2)mr²×v²/r² = mv²
= 100 × (0.20)²
= 100 × 0.04 = 4 joule