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)`.

Options

A. ` mg v_0 t cos theta hat k`
B. `-1/2 mg v_0 t^2 cos theta hat k`
C. ` 1/2 mg v_0 t^2 cos theta hat i`
D. ` - mg v_0 t^2 cos theta hat j`

Answer

B

Explanation

Let the particle projectied from (O) reach (P) in time (t) with velocity ` vec v = ( v_x hat I + hat I + v_y hat j) ` and corrdinates of (P) be ( x, y)
.
Then ` v_x = v_0 cos theta `,v_y = v_0 sin theta - gt`
` x = v_0 cos thetat, y= v_0 isn theta t - 1/2 gt^@`
`vec r = ( a hat i+ y hat j)`
Momentum of particle at (P) about origin (O) is `.
` vec L = vec r xx vec p = ( x hat i + y hat j ) xx m ( v_x hat i + v _y hat j)`
` = m y v_x (-hat k) + m x v_y (hat k)`
` = m ( v_0 sin theta t 1/2 gt^2) v_0 cos theta (- hat k)`
` + m ( v_0 cos theta t) ( v_0 sin theta - gt ) (hat k)
` =- mv_0^2 cos theta sin t hat k + me_0 cos theta xx 1/2 gt^2 hat k`
`+ mv_0^2 sin theta cos theta cos thea t hat k - mv-0 cos theta gt^2 hat k`
` =- 1/2 m v_0 gt^2 cos theta hat k` .