A yo-yo is made from two uniform disks each with mass m = 0.44 kg and radius R =

Question

A yo-yo is made from two uniform disks each with mass m = 0.44 kg and radius R = 19.9 cm, connected by a light axle of radius b = 2.0 cm. A light, thin string is wound several times around the axle and then held stationary while the yo-yo is released from rest, dropping as the string unwinds. Find the tension in the string just before the yo-yo reaches its lowest point.

Give your answer in Newtons to the first decimal place.

Note: You do not need to know how long the string is to solve this problem. There is a relation between linear acceleration and Tension, and a relation between angular acceleration and Tension that should allow you to solve this problem in terms of what you already know.

Explanation / Answer

The mass of the disk m = 0.44 kg The radius of disk R = 19.9 cm = 0.199 m The radius of a light axle  b = 2.0 cm = 0.02 m Then the moment of inertia of the system                I = (2m)R2/2 = mR2
The net force                F = (2m)g - T = 2ma   .................. (1)
Now the force acting on the pulley is torque,       = I          Tb   = I        Tb = mR2(a/b)        ma = Tb2/R2   ....................... (2)
From equations (2) and (1) , we get        (2m)g - T = 2Tb2/R2
therefore tension                            T = 2mg/(1 +2b2/R2)                                = [(2)(0.44 kg)(9.8 m/s2)] / [(1) + (2)(0.02 m / 0.199 m)2]                                = 8.45 N The radius of disk R = 19.9 cm = 0.199 m The radius of a light axle  b = 2.0 cm = 0.02 m Then the moment of inertia of the system                I = (2m)R2/2 = mR2
The net force                F = (2m)g - T = 2ma   .................. (1)
Now the force acting on the pulley is torque,       = I          Tb   = I        Tb = mR2(a/b)        ma = Tb2/R2   ....................... (2)
From equations (2) and (1) , we get        (2m)g - T = 2Tb2/R2
therefore tension                            T = 2mg/(1 +2b2/R2)                                = [(2)(0.44 kg)(9.8 m/s2)] / [(1) + (2)(0.02 m / 0.199 m)2]                                = 8.45 N

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