A yo-yo is a spool made up of two outer disks connected together by a small axle

Question

A yo-yo is a spool made up of two outer disks connected together by a small axle wrapped in string. The yo-yo that we will consider has disks of radius 3 cm and mass 0.06 kg and an axle of radius 1 cm and negligible mass. If you were to release the yo-yo and pull upward on the end of the string, with what minimum acceleration must you give the string in order for the yo-yo not to fall? If you were to place the yo-yo upright on the table and pull on its string, you would notice that it either rolls to the left or right, depending on the angle at which you apply the tension relative to the horizontal. What is the angle at which the yo-yo changes the direction in which it rolls?

Explanation / Answer

given

r = 0.03 m

m = 0.06 kg

ra= 1 cm

a. minimum upward acceleration = a

T - mg = ma

T*ra = I*a/ra

I = 0.5mr^2

T = 0.5mr^2*a/ra^2

0.5mr^2*a/ra^2 - mg = ma

a = g/(0.5r^2/ra^2 - 1)

a = 2.80285 m/s/s

b. let angle with the horizontal be theta

then

T*ra - f*r = 0 ( acceleration a = 0 at the pint where direction of net torque changes)

also

Tsin(theta) = mg

Tcos(theta) = f

hence

T*ra - Tcos(theta)*r = 0

ra/r = cos(theta)

theta = 70.528 deg

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