A spherical shell of radius 3.59 cm and a cylinder of radius 7.22 cm are rolling without slipping along the same floor. The two objects have the same mass. If they are to have the same total kinetic energy, what should the ratio of the spherical shell's angular speed to the cylinder's angular speed be?

(ω₁ / ω₂) = 1.9079

Explanation:

Given

R₁ = 3.59 cm

R₂ = 7.22 cm

m₁ = m₂ = m

K₁ = K₂

We know that

K₁ = Kt₁ + Kr₁ = 0.5*m₁*v₁²+0.5*I₁*ω₁²

if

v₁ = ω₁*R₁

and

I₁ = (2/3) * m₁*R₁² = (2/3) * m*R₁²

∴ K₁ = 0.5*m*ω₁²*R₁²+0.5 * (2/3) * m*R₁²*ω₁² (I)

then

K₂ = Kt₂ + Kr₂ = 0.5*m₂*v₂²+0.5*I₂*ω₂²

if

v₂ = ω₂*R₂

and

I₂ = 0.5*m₂*R₂² = 0.5*m*R₂²

∴ K₂ = 0.5*m*ω₂²*R₂²+0.5 * (0.5*m*R₂²) * ω₂² (II)

∵ K₁ = K₂

⇒ 0.5*m*ω₁²*R₁²+0.5 * (2/3) * m*R₁²*ω₁² = 0.5*m*ω₂²*R₂²+0.5 * (0.5*m*R₂²) * ω₂²

⇒ ω₁²*R₁² + (2/3) * R₁²*ω₁² = ω₂²*R₂²+0.5*R₂²*ω₂²

⇒ (5/3) * ω₁²*R₁² = (3/2) * ω₂²*R₂²

⇒ (ω₁ / ω₂) ² = (3/2) * R₂² / ((5/3) * R₁²)

⇒ (ω₁ / ω₂) ² = (9/10) * (7.22 / 3.59) ²

⇒ (ω₁ / ω₂) = (7.22 / 3.59) √ (9/10)

⇒ (ω₁ / ω₂) = 1.9079