Answer:
The time is 16 min.
Explanation:
Given that,
Time = 120 sec
We need to calculate the moment of inertia
Using formula of moment of inertia
![I=\dfrac{1}{2}MR^2](https://tex.z-dn.net/?f=I%3D%5Cdfrac%7B1%7D%7B2%7DMR%5E2)
If the disk had twice the radius and twice the mass
The new moment of inertia
![I'=\dfrac{1}{2}\times2M\times(2R)^2](https://tex.z-dn.net/?f=I%27%3D%5Cdfrac%7B1%7D%7B2%7D%5Ctimes2M%5Ctimes%282R%29%5E2)
![I'=8I](https://tex.z-dn.net/?f=I%27%3D8I)
We know,
The torque is
![\tau=F\times R](https://tex.z-dn.net/?f=%5Ctau%3DF%5Ctimes%20R)
We need to calculate the initial rotation acceleration
Using formula of acceleration
![\alpha=\dfrac{\tau}{I}](https://tex.z-dn.net/?f=%5Calpha%3D%5Cdfrac%7B%5Ctau%7D%7BI%7D)
Put the value in to the formula
![\alpha=\dfrac{F\times R}{\dfrac{1}{2}MR^2}](https://tex.z-dn.net/?f=%5Calpha%3D%5Cdfrac%7BF%5Ctimes%20R%7D%7B%5Cdfrac%7B1%7D%7B2%7DMR%5E2%7D)
![\alpha=\dfrac{2F}{MR}](https://tex.z-dn.net/?f=%5Calpha%3D%5Cdfrac%7B2F%7D%7BMR%7D)
We need to calculate the new rotation acceleration
Using formula of acceleration
![\alpha'=\dfrac{\tau}{I'}](https://tex.z-dn.net/?f=%5Calpha%27%3D%5Cdfrac%7B%5Ctau%7D%7BI%27%7D)
Put the value in to the formula
![\alpha=\dfrac{F\times R}{8\times\dfrac{1}{2}MR^2}](https://tex.z-dn.net/?f=%5Calpha%3D%5Cdfrac%7BF%5Ctimes%20R%7D%7B8%5Ctimes%5Cdfrac%7B1%7D%7B2%7DMR%5E2%7D)
![\alpha=\dfrac{2F}{8MR}](https://tex.z-dn.net/?f=%5Calpha%3D%5Cdfrac%7B2F%7D%7B8MR%7D)
![\alpha=\dfrac{\alpha}{8}](https://tex.z-dn.net/?f=%5Calpha%3D%5Cdfrac%7B%5Calpha%7D%7B8%7D)
Rotation speed is same.
We need to calculate the time
Using formula angular velocity
![\Omega=\omega'](https://tex.z-dn.net/?f=%5COmega%3D%5Comega%27)
![\alpha\time t=\alpha'\times t'](https://tex.z-dn.net/?f=%5Calpha%5Ctime%20t%3D%5Calpha%27%5Ctimes%20t%27)
Put the value into the formula
![\alpha\times120=\dfrac{\alpha}{8}\times t'](https://tex.z-dn.net/?f=%5Calpha%5Ctimes120%3D%5Cdfrac%7B%5Calpha%7D%7B8%7D%5Ctimes%20t%27)
![t'=960\ sec](https://tex.z-dn.net/?f=t%27%3D960%5C%20sec)
![t'=16\ min](https://tex.z-dn.net/?f=t%27%3D16%5C%20min)
Hence, The time is 16 min.