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A small particle of mass m is projected ...

A small particle of mass `m` is projected at an angle `theta` with x-axis with initial velocity `upsilon_(0)` in x-y plane as shown in Fig. Calculate the angular momentum of the particle
at `t lt (upsilon_(0) sin theta)/(g)`.

A

`mg upsilon_(0) t cos theta hatk`

B

`-(1)/(2) mg upsilon_(0) t^(2) cos theta hatk`

C

`(1)/(2)mg upsilon_(0) t^(2) cos theta hati`

D

`-mg upsilon_(0) t^(2) cos theta hatj`

Text Solution

Verified by Experts

The correct Answer is:
B
In Fig. the particle projected from `O`, reaches `P (x,y)` in `t` sec. with velocity
`overset rarr(upsilon) = (upsilon_(x)hati + upsilon_(y) hatj)`.
`:. upsilon_(x) = upsilon_(0) cos theta, upsilon_(y) = upsilon_(0) sin theta - g t`
and `x = (upsilon_(0) cos theta)t, y = (upsilon_(0) sin theta)t - (1)/(2)g t^(2)`
`overset rarr(r) = (x hati + y hatj)`
Moment of particle at `P`
`= m overset rarr(upsilon) = m(upsilon_(x) hati + upsilon_(y) hatj)`
= my `upsilon_(x) = (-hatk) + mx upsilon_(y) (hatk)`
`= m ((upsilon_(0) sin theta) t - (1)/(2)g t^(2)) upsilon_(0) cos theta (-hatk)`
`+ m(upsilon_(0) cos theta) r (upsilon_(0) sin theta - g t) (hatk)`
`= (-m upsilon_(0)^(2) cos theta sin theta) t hatk + m upsilon_(0) cos theta xx (1)/(2) g t^(2) hatk`
`+ (m upsilon_(0)^(2) sin theta cos theta) t hatk - m upsilon_(0) cos theta g t^(2) hatk`
`= - (1)/(2) m upsilon_(0) g t^(2) cos theta hatk`
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