Two circular discs of radius each 10 cm are joined at their centres by a rod of length 30 cm and mass 600 gm as shown in figure.
If the mass of each disc is 600 gm and applied torque between two discs is $43\times10^5$ dyne.cm, the angular acceleration of the discs about the given axis $AB$ is ______ $\text{rad/s}^2$.
Answer: (D) 11
The axis $AB$ is perpendicular to the rod, $10$ cm from the left disc's centre and $20$ cm from the right disc's centre. Each disc's plane is perpendicular to the rod, so $AB$ is parallel to a diameter of each disc. Use CGS units ($M = 600$ g, $R = 10$ cm).
Disc about its diameter: $\frac14MR^2 = \frac14(600)(100) = 15000$ g·cm².
Left disc: $15000 + 600(10)^2 = 75000$ g·cm²
Right disc: $15000 + 600(20)^2 = 255000$ g·cm²
Rod ($L = 30$ cm, centre at $5$ cm from $AB$): $\frac{1}{12}(600)(30)^2 + 600(5)^2 = 45000 + 15000 = 60000$ g·cm²
$$I = 75000 + 255000 + 60000 = 3.9\times10^5\ \text{g·cm}^2$$
$$\alpha = \frac{\tau}{I} = \frac{43\times10^5}{3.9\times10^5} \approx 11\ \text{rad/s}^2$$
Solution by Sreeraj P, M.Sc Physics