A cylinder of radius 2.84 cm 2.84 cm and a spherical shell of radius 6.47 cm 6.47 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 cylinder's angular speed to the spherical shell's angular speed be?

2.4

Explanation:

cylinder moment of inertia = Iₔ = ½mrₔ² = ½2.84²

spherical shell moment of Iₑ = (2/3)mrₑ² = (2/3)6.47²

let the cylinder's angular speed be ωₔ and the spherical shell angular speed = ωₑ

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

for cylinder = ½Iₔωₔ² + ½mωₔ²rₔ² > [vₔ = rₔωₔ]

for sphere = ½Iₑωₑ² + ½mωₑ²rₑ²

=> ratio = 1

=> ωₔ²[½2.84² + 2.84²] / {ωₑ²[(2/3)6.47² + 6.47²]} = 1

ωₔ²(12.0894) = ωₑ²(69.7682)

(ωₔ/ωₑ)² = [69.7682] / [12.0894] ~= 5.77

=> (ωₔ/ωₑ) = √(5.77) = 2.4

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now this weight will have two components one is along the incline other is perpendicular to it

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so here the component of weight is given as

here we have

so it requires 75 n force to move it upwards