Consider a body of mass 1.0 at rest at the origin at time `t = 0`. A force `vec F=(alpha t hat l +beta hat J) ` is applied on the body, where `alpha =1.0 Ns^(-1) and beta =1.0 N`. The torque acting on the body about the origin at time `t=1.0 s " is " vec tau` . Which of the following statements is (are) true?

Options

A. `|tau|=(1)/(3)Nm`
B. The torque is in the direction of the unit vector `+hatk`
C. The velocity of the body at t=1 s is `vecv=(1)/(2)(hati+2hatj) ms^(-1)`
D. The magnitude of displacement of the body at `t=1 s is `(1)/(6)` m.

Answer

A, C

Explanation

`m=1kg`
`m vaca= vecF`
`1(veca)= thati+hatj`
`rArr aeca= t hati+ hatj`
`rArr (d vecv)/(dt)= t hati+ hatj`
`rArr vec=(t^(2))/(2) hati+ t hatj " " ...(i)`
Substituting t - 1 in equation (i), we can find velocity of the particle at t=1.
`rArr vec v=(1)/(2) hati+ hatj=(1)/(2) (hati +2hatj)`
Hence option (c) is correct.
Equation (i) can be written as follows:
`rrr (d vecs)/(dt)=(t^(2))/(2) hati+thatj`
`rArr vec s=(t^(2))/(8) hati+(t^(2))/(2)hatj`
`rarr vec s(t=1) (1)/(6) hati+(1)/(2) hatj`
This is same as position vector `vecr` of the particle at t=1s.
`vecF= t hati+ hatj= hati+ hatj`
`vecv=(1)/(6) hati+(1)/(2) hatj`
`tau= vecr xx vecF`
`tau=[(1)/(6) hati+(1)/(2) hatj] xx [hati+hatj]`
`=(1)/(6) (hatixx hatj)+(1)/(2) (hatjxx hati)`
`=(1)/(6) hatk-(1)/(2) hatk= (-2)/(6) hatk=-(1)/(3) hatk`
`|vectau|=(1)/(3)at t=1 sec`
Hence option (c is correct.