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The position vector vacr of the z-compon...

The position vector `vacr` of the z-component of `vecL` is `55ma^2omega`. Following equation
`vacr(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?

A

The velocity `vacv` is given by `vacv = (10hati+10hatj) ms^(-1)`

B

The angular momentum `vacL` with respect to the origin is given by `vacL = -5/3 hatk N ms.`

C

The force `vacF` is given by `vacF = (hati+2hatj)N`

D

The torque `vactau` with respect to the origin is given by `vactau = -(20/3) hatk Nm`

Text Solution

Verified by Experts

The correct Answer is:
A, B
(a,b)
`vecr = alpha t^3 + betat^2 hatj`
`vecr = (10)/(3) t^3 hati + 5t^2 hatjm`
`vecv = (dvecr)/(dt) = 10t^2 hati + 10thatj ms^(-1)`
`vecalpha = (dvecv)/(dt) = 20t hatj + 10hatjms^(-2)`
At t = 1s
`vecr_(t=1) = (10)/(3)hat I + 5hatjm`
`vecv_(t =1) = 10hati + 10hatj ms^(-1)`
`hatP_(t =1) = hati + hatj kgms^(-1)`
`alpha_(t =1) = 20hati + 10 hatj ms^(-2)`
`vecL = vecrxxvecP = |(hat i,hat j,hat k ),(10/3,5,0),(1,1,0)| = hatk[(10)/(3) -5]=-5/3 hatk kgms^(-1)`
`vecF = mveca = (2hati + hatj)N`
`vectau = vecrxxvecF=|(hat i,hat j,hat k ),(10/3,5,0),(2,1,0)|=hatk[(10)/(3) -10]=(-20)/(3) hatkN-m.`
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