A small disk of radius R₁ is fastened coaxially to a larger disk of radius R₂.The combination is free to rotate on a fixed axle, which is perpendicular to a horizontal frictionless table top, as shown in the overhead view below.The rotational inertia of the combination is I.A string is wrapped around the larger disk and attached to a block of mass m, on the table.Another string is wrapped around the smaller disk and is pulled with a force as shown.The tension in the string pulling the block is: A)R₁F/R₂ B)mR₁R₂F/(I - mR₂²) C)mR₁R₂F/(I + mR₂²) D)mR₁R₂F/(I - mR₁R₂) E)mR₁R₂F/(I + mR₁R₂)

Question

Quizplus · Accepted Answer

The system has two components: the small disk and the large disk with the attached block. The tension in the string pulling the block is equal to the force exerted on the small disk since the strings are assumed to be ideal and inextensible. 
The force exerted on the small disk produces a torque given by τ=F(R1), where R1 is the radius of the small disk. The tension force on the larger disk produces a torque equal to τ=F(R2), where R2 is the radius of the larger disk. 
The rotational inertia of the system is given by I=1/2m(R1^2+R2^2), where m is the mass of the larger disk and block. 
Now, applying Newton's second law for rotational motion, we have:
τnet=Iα
where τnet is the net torque acting on the system and α is its angular acceleration. 
The net torque on the system is given by:
τnet=F(R2-R1)
since the two torques due to the tension forces act in opposite directions. 
We can now solve for the tension in the string pulling the block:
F=(Iα)/(R2-R1)
α=(F(R2-R1))/I
The angular acceleration of the system is the same for both disks since they are rigidly connected. Thus, we can use this equation to solve for the tension in the string pulling the block:
Tension force on block = F = (Iα)/(R2-R1) = (mR1R2F)/(I + mR1^2)
Therefore, the tension in the string pulling the block is given by:

Tension force on block = m*R1*R2*F / (I + m*R1^2) 
which is closest to option C.

A Small Disk of Radius R1 Is Fastened Coaxially to a Larger

A Small Disk of Radius R₁ Is Fastened Coaxially to a Larger