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29 August, 09:40

A sphere of radius 1.59 cm and a spherical shell of radius 7.72 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 sphere's angular speed to the spherical shell's angular speed be?

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  1. 29 August, 12:25
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    Answer: 4.86

    Explanation:

    sphere moment of Inertia Iₑ = (2/5) mrₑ²

    Let the sphere of radius 1.59 cm be x

    Let the spherical shell of radius 7.72 cm be y, so that

    Iₑ (x) = 2/5 * m * 1.59²

    Iₑ (x) = 2/5 * m * 2.5281

    Iₑ (x) = 1.011m

    Iₑ (y) = 2/5 * m * 7.72²

    Iₑ (y) = 2/5 * m * 59.5984

    Iₑ (y) = 23.84m

    Also, the angular speed of the sphere's would be ωₑ (x) and ωₑ (y)

    total k. e = rotational k. e + linear k. e

    for sphere = ½Iₑωₑ² + ½mωₑ²rₑ²

    For sphere x

    {ωₑ²[ 1.011 + 1.59²]} =

    ωₑ² (1.011 + 2.5281) =

    ωₑ² (3.5391)

    For sphere y

    {ωₑ²[ 23.84 + 7.72²]} =

    ωₑ² (23.84 + 59.5984) =

    ωₑ² (83.4384)

    If the ratio of x/y = 1, then

    ωₑ (x) ² (3.5391) / ωₑ (y) ² (83.4384) = 1

    ωₑ (x) ² (3.5391) = ωₑ (y) ² (83.4384)

    [ωₔ (x) / ωₑ (y) ]² = [83.4384] / [3.5391] ~ = 23.5762

    [ωₔ (x) / ωₑ (y) ] = √ (23.5762)

    [ωₔ (x) / ωₑ (y) ] = 4.86
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