E. General Physics - Give complete, neatly done and correct solution done on com

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E. General Physics - Give complete, neatly done and correct solution done on computer for clarity with justified and complete explanation of proofs. Emphasis on complete and correct solution. The answers are verified with the solution but need clear proofs before rating. This is all the information provided. Thank you very much.

25. A super Yo-Yo is constructed with a heavy rim supported by light spokes around an axle whose diameter is one quarter that of the rim. A 1 m long string is wound around the axle. The end of the string is held fixed and the Yo-Yo allowed to fall. What is the maximum downwards speed of the Yo-Yo?

Explanation / Answer

25 from the given data

length of string , l = 1 m

let radius of yo-yo be R, radius of axle be r

mass of yo you be m, anbd moment of inertia then becomes 0.5mR^2

so at any point of time t

tension in the string be T

then

from force balance

mg - T = ma ( where a is acceletration of yo yo)

from torque balance

Tr = I*alpha = 0.5mR^2*alpha

but

alpha*r = a

hence

Tr = 0.5mR^2*a/r

T = 0.5mR^2*a/r^2

hence

mg - 0.5mR^2*a/r^2 = ma

a = g/(1 + 0.5R^2/r^2)

hence acceleration is constant

so maximum downwards speed = v

2*a*l = v^2

v = sqrt(2g/(1 + R^2/2r^2))

this is maximum speed of yo yo, where r is radius of axle , R is radius of yo yo and g as acceleration due to gravity

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