A yo-yo is made of three disks, as shown : two outer disks of mass M = 40 g, rad

Question

A yo-yo is made of three disks, as shown : two outer disks of mass M = 40 g, radius R = 3.0 cm, and thickness d = 1.2 cm, sandwiched around an inner disk of mass m = 20 g, radius r = 1.5 cm, and the same thickness d = 1.2 cm. A string is wrapped around the inner disk. The free end is attached to a ceiling, and then the yo-yo is released from rest.

a) What is the tension in the string as the yo-yo falls? What fraction of the yo-yo's weight is this?

b) What is the yo-yo's linear speed when it has fallen a distance of H = 0.5 m? How does this compare to what the speed would be if it fell with no string attached?

Explanation / Answer

b)ma = mg - T

but ; T r = I alpha ; alpha = a/R

T = (I/r^2)a = (m R^2 /2r^2 ) a

ma = mg - (m R^2 /2r^2 ) a

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

T = mg/[1 + 2r^2/R^2]

T = 0.04 x 9.8 / [1 + 2 x 0.015^2/0.03^3] = 0.02 N

Hence, T = 0.02 N

B)From energy conservation

1/2 m v^2 + 1/2 I omega^2= m g h

but omega = v/R and I = 1/2 mR^2

3/4 mv^2 = m g h

v = sqrt (4 g h/3)

v = sqrt(4 x 9.8 x 0.5/3) = 2.56 m/s

Hence, v = 2.56 m/s

If the yo-yo is not attached to any string, it would have fallen freely like any other object under the influence of gravitaional force. In that case the magnitude of velocity will be

v' = sqrt(2 g h) = sqrt (2 x 9.8 x 0.5) = 3.13 m/s

Hence, v' > v

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