A small particle of mass m is projected at an angle θ with the x-axis with an initial velocity v0 in the x-y plane as shown in the figure. At a time t

Question

A small particle of mass m is projected at an angle θ with the x-axis with an initial velocity v0 in the x-y plane as shown in the figure. At a time t < (v0 sin θ/g), the angular momentum of the particle is where hati, hatj and hatk are unit vectors along x, y and z-axis respectively. (A) -mgv0 t2 cos θ hatj (B) -mgv0 t cos θ hatk (C) -(1/2)mgv0 t2 cos θ hatk (D) (1/2)mgv0 t2 cos θ hati

Tardigrade Admin · Accepted Answer

vecL = m( vecr× vecv) vecL = m[v0 cos θ t hati +(v0 sin θ t-(1/2)gt2) hatj]×[v0 cos θ hati+(v0 sin θ-gt) hatj] = mv0 cos θ t[-(1/2)gt] hatk -(1/2)mgv0 t2 cos θ hatk

$- m g v_{0} t^{2} cos θ \hat{j}$

$- m g v_{0} t cos θ \hat{k}$

$- \frac{1}{2} m g v_{0} t^{2} cos θ \hat{k}$

$\frac{1}{2} m g v_{0} t^{2} cos θ \hat{i}$

$L = m (r \times v)$
$L = m [v_{0} cos θt \hat{i} + (v_{0} s in θt - \frac{1}{2} g t^{2}) \hat{j}] \times [v_{0} cos θ \hat{i} + (v_{0} s in θ - g t) \hat{j}]$
$= m v_{0} cos θt [- \frac{1}{2} g t] \hat{k}$
$- \frac{1}{2} m g v_{0} t^{2} cos θ \hat{k}$

Q. A small particle of mass m is projected at an angle θ with the x-axis with an initial velocity v0​ in the x−y plane as shown in the figure. At a time t<gv0​sinθ​, the angular momentum of the particle is where i^,j^​ and k^ are unit vectors along x,y and z−axis respectively.

−mgv0​t2cosθj^​

−mgv0​tcosθk^

−21​mgv0​t2cosθk^

21​mgv0​t2cosθi^

L=m(r×v) L=m[v0​cosθti^+(v0​sinθt−21​gt2)j^​]×[v0​cosθi^+(v0​sinθ−gt)j^​] =mv0​cosθt[−21​gt]k^ −21​mgv0​t2cosθk^

$- m g v_{0} t^{2} cos θ \hat{j}$

$- m g v_{0} t cos θ \hat{k}$

$- \frac{1}{2} m g v_{0} t^{2} cos θ \hat{k}$

$\frac{1}{2} m g v_{0} t^{2} cos θ \hat{i}$

$L = m (r \times v)$
$L = m [v_{0} cos θt \hat{i} + (v_{0} s in θt - \frac{1}{2} g t^{2}) \hat{j}] \times [v_{0} cos θ \hat{i} + (v_{0} s in θ - g t) \hat{j}]$
$= m v_{0} cos θt [- \frac{1}{2} g t] \hat{k}$
$- \frac{1}{2} m g v_{0} t^{2} cos θ \hat{k}$