Question

What is the ratio of disk velocity to hoop velocity (in case of rotational kinetic energy

Answer 1

Answer:

the ratio of disk velocity to hoop velocity (in case of rotational kinetic energy is 12+26272.282

Answer 2

Therefore the ratio of disc velocity to hoop velocity is 2/√3

Given:

Kinetic energy = KE = 1/2 mv²

Rotational Kinetic Energy = RKE = 1/2 Iω²

Where m = Mass of the body

v = velocity of the body

I = Moment of Inertia in the body

ω = Angular velocity of the body

Let Disc = 1 and Hoop = 2

Moment of Inertia of Disc = I₁ = 1/2 m₁r₁²

Moment of Inertia of Hoop = I₂ = m₂r₂²

To Find:

The ratio of the disc velocity to the hoop velocity.

Solution:

This problem can be solved in this easy method as shown below.

Angular velocity ω = ( Linear velocity v ) / Radius r ⇒ ω = v / r

Now Rotational Kinetic Energy of Disc = RKE₁ = 1/2 I₁ω₁²

⇒  Rotational Kinetic Energy of Disc = RKE₁ = 1/2 ( 1/2 m₁r₁² ) ( v₁/r₁ )²    { ∵ I₁ =1/2 m₁r₁² and ω₁ = v₁/r₁ }

⇒  Rotational Kinetic Energy of Disc = RKE₁ = 1/4 m₁v₁²

Now Rotational Kinetic Energy of Hoop = RKE₂ = 1/2 I₂ω₂²

⇒  Rotational Kinetic Energy of Hoop = RKE₂ = ( 1/2 m₂r₂² ) ( v₂/r₂ )²    { ∵ I₂ = m₂r₂² and ω₂ = v₂/r₂ }

⇒  Rotational Kinetic Energy of Hoop = RKE₂ = 1/2 m₂v₂²

Now let us assume that both the disc and the hoop are made to roll over an inclined plane at the same height 'h'.

So total Potential Energy of the body is converted to Kinetic Energy.

For Disc:

Potential Energy = PE = m₁gh

g = Acceleration due to gravity

Translational Kinetic Energy = TKE = 1/2 m₁v₁²

Rotational Kinetic Energy = RKE = 1/4 m₁v₁²

By applying Energy Conservation, we get,

⇒ PE = TKE + RKE

⇒ m₁gh = 1/2 m₁v₁² + 1/4 m₁v₁²

⇒ m₁gh = 3/4 m₁v₁² ⇒ 3/4 gh = v₁² ⇒ v₁ = √(4gh/3)

Hence the velocity of the disc = v₁ = √(4gh/3)

For Hoop:

Potential Energy = PE = m₂gh

g = Acceleration due to gravity

Translational Kinetic Energy = TKE = 1/2 m₂v₂²

Rotational Kinetic Energy = RKE = 1/2 m₂v₂²

By applying Energy Conservation, we get,

⇒ PE = TKE + RKE

⇒ m₂gh = 1/2 m₂v₂² + 1/2 m₂v₂²

⇒ m₂gh = m₂v₂² ⇒ gh = v₂² ⇒ v₂ = √(gh)

Hence the velocity of the Hoop = v₂ = √(gh)

Now the ratio of disc velocity to hoop velocity is given by,'

⇒ v₁/v₂ =√(4/3gh) /√(gh) ⇒ v₁/v₂ = √(4/3 )

Therefore the ratio of disc velocity to hoop velocity is 2/√3.

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