Question

2R 2R m M L 2ri (14 HIHIHIHIHIHIHIHIHIHIHHHHHH 2. A cylinder with length 1 (along the z direction), uniform mass density p (kg/m3]) be- tween radii r and R >r has total mass m and is placed on the flat bed of a truck with mass M, as indicated in the above side view. At t = 0 the truck is at rest and the cylinder is at the very front of the truck bed. Static and kinetic friction coefficients between cylinder and truck's surface are us > Uk > 0. Fort > 0 the truck has thrust T> 0 in the positive direction. a. Express the mass m and the moment of inertia I around the center of mass of the cylinder in terms of p, r, R, and l. Also express I in terms of m, R, and r. You can now consider m and I known. b. Write the relevant equations of motion for the translation (x), rotation (0) of the cylinder, and the motion of the truck for as long as the cylinder remains on the truck. Assume perfect rolling of the cylinder on the surface. c. Determine the minimum value of the static friction such that the assumption of perfect rolling is true. d. Write the relevant equations of motion for the translation (x), rotation () of the cylinder and the motion of the truck for as long as the cylinder remains on the truck. Assume sliding at the interface between the cylinder and the surface of the truck.

Answer 1

A x mm Given date :- p length mass inner outer Mass of Mass of of cylinder al density of cylinder = radius " " or Radius " " R cylinder =m truck M. Asked :-(a) m & I in l. terms of p,r.R, & Solution :- mass as we know :- den density - volume over volume - Cross Sectional area x Saeight cross sectional area of Cylinder = (R2-72) - height / length of cylinder = 1 Volume of cylinder - (R?- x2) xe v = al (R²-22) sa im m= pv (5-8 smagu density) is mass of Cylinder (m) - parece-8?)|

Now af moment of ineacha x mass x (o'+R) ( hollow Cylinder the given object is a hollow cylinder af length l k inner a outer radii as r&R Respectively. ? L T I = {xm *(3*+ R") I = à x [ Pat(R2-74)] *(7°+R) şim - calculated earlier) 1 - Pore (R-32) terms of m , R & r . TIEŽ m * (r’+R) I in