The position vector `vecr` of a particle of mass m is given by the following equation
`vecr(t)= alphat^3hati+beta^2hatj,`
where `alpha = 10//3ms^(-3), beta = 5 ms^(-2)` and m = 0.1kg. At t=1s, which of the following statement (s) is (are) true about the particle?

Options

A. The velocity ` vec v` is given by ` vec v = ( 10 hat i+ 10 hat j) ms^(-1)`
B. The angular momentum ` vec L` with respect to the origin is given by ` vec L =- (5// 3) hat k N ms`
C. The force ` vec F` is given by ` vec F = ( hat i+ 2 hat j) N`
D. The torque with respect to the origin is given by ` vec tou =- (20)/3 hat k Nm`

Answer

A, B, D

Explanation

` vec t= alpha t^3 + beta t^2 hat j` ltBrgt velocity, ` vec = (vec (dr) /(dt) = 3 alpha t^2 hat I +2 beta t hat j`
Vcceleration, ` vec a= (vec (dv))/(dt) = 6 alpha t hat i+ 2 beta hat j`
At ` t= 1 s`,
(a) ` vec v = 3 xx (10)/3 xx (1)^2 hat I + 2 xx 5 xx 1 xx hat j`
`= (10 hat i = 10 hat j) ms^(-1)`
(b) Angular momentum ` hat L = vec r xx vrc p = vec r xx m vec v`
:. ` vec L = [ (10)/3 (1)^3 hat i + 5 (1)^2 hat j] xx 0.1 (10 hat i + 10 haat j)`
`= ( - 5/3 hat k) N ms`
(c ) Force ` vec F= m vec a =0.1 ( 6 xx (10)/3 xx hat i + 2 aht 5 hat j)`
`= (2 hat i +hat j) N`
(d) Torque ` vec tau = vec r xx vec F`
`= [ (10)/3 (1)^3 aht i +5 (1)^2 hat j ] xx (2 hat i + hat j)`
`= ((10)/3 hat i +5 hat j) xx (2 hat i + hat j)`
`= (10)/3 hat k + 10 (- hat k) = (- (20)/3 hat k) N-m`