Two thin circular disks of mass 2kg 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 disk through their centres. This object is kept on a truck in such a way that the axis of the object is horizontal and perpendicular to the direction of the 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 `9m//s^2` Calculate:
(i) The force fo friction on each disk,
(ii) 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 in terms of unit vectors `hati, hatj and hatk` in the x,y, and z directions.

Answer

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

Explanation

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`.