The position vector `vec(r)` of a particle of mass m is given by the following equation `vec(r) (t) = at^(3) hat(i) + beta t ^(2) hat(j) " where " alpha = 10//3 ms^(-3),beta = 5 ms^(-2) and m = 0 . 1 kg ` . At t = 1 s, which of the following statement (s) is (are) true about the particle ?

Options

A. The velocity `vecv` is given by `vecr = (10hati + 10 hatj) ms^(-1)`
B. The angular momentum `vecL` with respect to the origin is given by `vecz=-(5//3)hatk N m s`
C. The force `vecF` is given by `vecF = (hati + 2hatj)N`
D. The torque with respect to the origin is given by `tau=(-20//3)hatkNm`

Answer

A, B, D

Explanation

`vecr = alphat^(2) hati + beta t^(2)hatj`
`x=at^(3), y=beta t^(2)`
`vecr(1) =10/3 hati + 5hatj`
`(dx)/(dt) =3t^(2) alpha =10 rArr (dy)/(dt) = 2t beta =10`
`a_(N) =6alpha t = 10/3 xx 6 xx 1 =20, a_(Y) = 2beta`
(A) `V=(10hati + 10 hatj)` True.
(B) `vecL = m(vecr xx vecv),`................ `vecL =0.1 (10/3 hati + 5hatj) xx (1) hati + 10 hatj), vecL = -5/3 hatk` True.
( C) `vecF = 0.1 xx (20 hati + 10hatj) = (2hati + hatj)` Not True.
( D) `vectau = vecr xx vecF`, `vectau =(10/3 hati + 50 hatj) xx (2hat i + hatj), vectau =-20/3 hatk` True.