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

we need answer choices

when jerome paddle in flowing river towards east the net speed of jerome will increase

so the net speed of jerome will be given as

now the speed of paul will be same as it is given as he is running on the ground

so

so here we can say since the speed of jerome is more so he will win the race

now as we know the distance so the time difference is given as

so jerome will reach first and the time difference will be 203.55 s or 3.4 min