Q. Portion AB of the wedge shown in figure is rough and BC is smooth.A solid cylinder rolls without slipping from A to B.The ratio of translational kinetic energy to rotational kinetic energy,when the cylinder reaches point C is:-
The initial vertical height of point A is `H` . Point B lies at the height `H/2` .
Between A and B the variation in potential energy transforms into rotational energy + kinetic energy because there are friction forces that place in rotation the cylinder. The speed of the cylinder at point B is `V_1` and its angular speed is `omega_1` .
`(I*omega_1^2)/2 +(m*V_1^2)/2 =m*g*H/2`
with the condition `V_1 =omega_1*R`
For a solid cylinder the momentum of inertia is
Thus the first relation becomes
`(m*R^2)/2 *(V_1/R)^2 +m*V_1^2 =m*g*H`
`(3/2)*V_1^2 =g*h` or equivalent `V_1^2 =2/3*g*H`
Between points B and C there is no friction to further increase the rotational speed of the cylinder. The variation of potential energy transforms only into kinetic energy. The speed of cylinder at point C is `V_2` and its angular speed stay unchanged `omega_1` .
`(m*V_2^2)/2 -(m*V_1^2)/2 = m*g*H/2`
`V_2^2 -V_1^2 =g*H`
`V_2^2 -2/3*g*H =g*H`
`V_2^2 =5/3*g*H = 5/2*V1^2`
Now we can compute the ratio of translational kinetic energy to rotational energy at point C:
`(m*V_2^2)/(I*omega_1^2) = (m*(5/2)*V_1^2)/((mR^2)/2*(V_1^2/R^2)) =5`
The correct answer is B) 5.