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A Circular Disc Of Mass ‘M’ And Radius ‘r’ Is Set Rolling On A Table. If ‘ω’ Be The Angular Velocity, Calculate Its Total Energy ‘E’.

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Ans.

When a circular disc of mass M and radius r rolls on a table, then its total kinetic energy E will be,

E = (Energy due to linear motion) + (energy due to rotational motion)

The moment of inertia of a circular disc about an axis passing through the centre of the disc and perpendicular to the plane of the disc is
I=\frac{1}{2}Mr^2
[ To know the derivation (CLICK HERE) ]

Energy due to the linear motion is E_l=\frac{1}{2}Mv^2\\=\frac{1}{2}M{(r\omega)}^2\\=\frac{1}{2}Mr^2{\omega}^2

Energy due to the rotational motion is E_r=\frac{1}{2}I{\omega}^2 , [ To know the derivation of rotational kinetic energy, (CLICK HERE) ]

Putting the value of I we get, E_r=\frac{1}{4}Mr^2{\omega}^2 .

So the total kinetic energy of the disc is

E = E_l+E_r\\=\frac{1}{2}Mr^2{\omega}^2+\frac{1}{4}Mr^2{\omega}^2\\=\frac{3}{4}Mr^2{\omega}^2

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