Question

A trolley initially at rest with a solid cylinder placed on its bed such that cylinder axis makes angle $\theta$ with direction of motion of trolley as shown inthe figure starts to move forward with constant acceleration $a$. If initial distance of mid-point of cylinder axis from rear edge of trolley bed is $d$, calculate the distance $s$ which the trolley goes before the cylinder rolls off the edge of its horizontal bed. Assume dimensions of cylinder to be very small in comparison to other dimensions. Neglect slipping.
Calculate also, frictional force acting on the cylinder.

Answer 1

$netacc.=(a\sin \theta -r\alpha )F_2=ma\cos \theta \therefore F_1=m(a\sin \theta -r\alpha )also{F}_{1}r=I\alpha \Rightarrow F_1=\frac{1}{2}mr\alpha alsor\alpha =\frac{2}{3}a\sin \theta s=dcosec\theta .......bydiagramofquestion\frac{1}{2}a{t}^2=dcosec\theta \Rightarrow t=\sqrt{\frac{3\alpha }{a{\sin }^2\theta }}{F}_1=\frac{1}{2}m\alpha =\frac{1}{3}ma\sin \theta netfriction=\sqrt{f_1^2+f_2^2}=\frac{1}{3}ma\sqrt{{\sin }^2\theta +9{\cos }^2\theta }$