Discuss the effect of rolling on inclined plane and derive the expression for the acceleration.

Explanation

Acceleration of the rolling object :
(i) From the figure, it is seen that the component of gravitatinal force (mg cos`theta`) perpendicular to inclined plane is balanced by the normal force N.
Therefore, the component of gravitational force (mg sin`theta`) parallel to inclined plane and the frictional force f, combinely causes the motion.
(iii) For translational motion,
`mg sin theta - f = ma" "...(1)`
(iv) For rotational motion, let us take a torque with respect to the center of the object. mg sin`theta` cannot make any torque as it passes through the center of the object, but the frictional force f can set a torque as,
`tau = Rf`
(v) But we know, `tau = I alpha`, thus,
`Rf = I alpha`
(vi) Substituting, the angular acceleration `alpha = (a)/(R)` and the moment of inertia `I = mK^(2)`, we get,
`Rf = mK^(2) (a)/(R)`
`f = ma ((K^(2))/(R^(2)))" "...(2)`
(vii) Substituting equation (2) in (1), we have,
`mg sin theta - ma ((K^(2))/(R^(2)))=ma`
`mg sin theta = ma + ma ((K^(2))/(R^(2)))`
`a (1+(K^(2))/(R^(2))) = g sin theta`
(vii) After rewriting it for acceleration, we get
`a = (g sin theta)/((1 + (K^(2))/(R^(2))))`