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25 May, 07:28

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?

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  1. 25 May, 10:07
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    (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
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