Two solid discs, A and B, are made of the same material. Disc A has radius R and thickness t . Disc B has radius 3R and thickness t 3 . Identical constant tangential forces F are applied to the rim of each disc, causing them to rotate about their respective central axes. If the ratio of the angular acceleration of disc A to that of disc B is x , then the value of x is ____.

Step-by-step solution

Let the density of the material be . The moment of inertia of a solid disc is I = 1 2 MR^2 = 1 2 ( R^2 t)R^2 = 1 2 R^4 t . For disc A: I_A = 1 2 R^4 t For disc B: I_B = 1 2 (3R)^4 ( t 3 ) = 1 2 (81R^4) ( t 3 ) = 27 ( 1 2 R^4 t ) = 27 I_A The torque produced by the tangential force F is = F radius . For disc A: _A = F R For disc B: _B = F (3R) = 3 _A Using Newton's second law for rotation, = I : For disc A: _A = _A I_A For disc B: _B = _B I_B = 3 _A 27 I_A = 1 9 _A The ratio of their angular accelerations is: _A _B