Question

Two thin circular disks of mass `2 kg` and radius `10 cm` each are joined by a rigid massless rod of length `20 cm`. The axis of the rod is along the perpendicular to the planes of the disks through their centre. The object is kept on a truck in such a way that the axis of the object is horizontal and perpendicular to the direction of motion of the truck. Its friction with the floor of the truck is large enough so that the object can roll on the truck without slipping. Take x-axis as the direction of motion of the truck and z-axis as the vertically upwards direction. If the truck has an acceleration of `9 m//s^(2)` calculate :
(a) the force of friction on each disk

(b) The magnitude and the direction of the frictional torque acting on each disk about the centre of mass `O` of the object. Express the torque in the vector form of unit vectors in the `x-y` and `z` directions.

Answer 1

Correct Answer - `6N, -0.6 hat j +- 0.6 hat k`

For translation motion
`F_(Psuedo) - 2f = (2m) a` …(1)
For rotational motion
`2f xx R = (2 I) alpha`
`f = (I)/(R) (a)/(R) rArr f = ((1//2)mR^(2))/(R^(2)) xx (a)/(R)`
`f = (1)/(ma)`...(2)
From (1) and (2)
`F_("Psuedo") - 2f = 4f rArr f = (2xx2xx9)/(6) rArr f = 6N`
`vec tau_(P) = vec tau_(P) xx vec f rArr (-0.1 hati - 0.1 hat k) xx (6 hat i)`
`vec tau_(P) = 0.6 hat k - 0.6 hat j rArr vec tau _(P) = 0.6 hat j + 0.6 hat k`
`vec tau_(Q) = (0.1 hatj - 0.6 hat k) xx (6 hat i)`
`vec T_(P) = -0.6 hat j - 0.6 hat k`.