- Home
- Standard 11
- Physics
Find the components along the $x, y, z$ axes of the angular momentum $l$ of a particle. whose position vector is $r$ with components $x, y, z$ and momentum is $p$ with components $p_{ r }, p_{ y }$ and $p_{z} .$ Show that if the particle moves only in the $x -y$ plane the angular momentum has only a $z-$component.
Solution
$l_{ x }=y p_{ z }-z p_{ y } l_{ y }$
$=z p_{ x }- x p_{ z } l_{ z }$
$=x p_{y}-y p_{x}$
Linear momentum of the particle, $\vec{p}=p_{ x } \hat{ i }+p_{y} \hat{ j }+p_{2} \hat{ k }$
Position vector of the particle, $\vec{r}=x \hat{ i }+y \hat{ j }+z \hat{ k }$
Angular momentum, $\vec{l}=\vec{r} \times \vec{p}$
$=(x \hat{ i }+y \hat{ j }+z \hat{ k }) \times\left(p_{x} \hat{ i }+p_{y} \hat{ j }+p_{z} \hat{ k }\right)$
$=\left|\begin{array}{ccc}\hat{ i } & \hat{ j } & \hat{ k } \\ x & y & z \\ p_{x} & p_{y} & p_{z}\end{array}\right|$
$l_{ x } \hat{ i }+l_{ y } \hat{ j }+l_{ z } \hat{ k }$$=\hat{ i }\left(y p_{z}-z p_{ y }\right)-\hat{ j }\left(x p_{z}-z p_{ x }\right)+\hat{ k }\left(x p_{y}-z p_{ x }\right)$
Comparing the coefficients of $\hat{ i }, \hat{ j },$ and $\hat{ k },$ we get:
$\left.\begin{array}{l}l_{ x }=y p_{ z }-z p_{ y } \\ l_{ y }=x p_{ z }-z p_{ x } \\ l_{z}=x p_{y}-y p_{ x }\end{array}\right\}$ $\dots(i)$
The particle moves in the $x$ $-y$ plane. Hence, the $z$ -component of the position vector
and linear momentum vector becomes zero, i.e., $z=p_{z}=0$
Thus, equation ( $i$ ) reduces to:
$l_{x}=0$
$l_{y}=0$
$l_{z}=x p_{y}-y p_{x}$
Therefore, when the particle is confined to move in the $x-y$ plane, the direction of angular momentum is along the $z$ -direction.
Similar Questions
In the List-$I$ below, four different paths of a particle are given as functions of time. In these functions, $\alpha$ and $\beta$ are positive constants of appropriate dimensions and $\alpha \neq \beta$. In each case, the force acting on the particle is either zero or conservative. In List-II, five physical quantities of the particle are mentioned: $\overrightarrow{ p }$ is the linear momentum, $\bar{L}$ is the angular momentum about the origin, $K$ is the kinetic energy, $U$ is the potential energy and $E$ is the total energy. Match each path in List-$I$ with those quantities in List-$II$, which are conserved for that path.
| List-$I$ | List-$II$ |
| $P$ $\dot{r}(t)=\alpha t \hat{t}+\beta t \hat{j}$ | $1$ $\overrightarrow{ p }$ |
| $Q$ $\dot{r}(t)=\alpha \cos \omega t \hat{i}+\beta \sin \omega t \hat{j}$ | $2$ $\overrightarrow{ L }$ |
| $R$ $\dot{r}(t)=\alpha(\cos \omega t \hat{i}+\sin \omega t \hat{j})$ | $3$ $K$ |
| $S$ $\dot{r}(t)=\alpha t \hat{i}+\frac{\beta}{2} t^2 \hat{j}$ | $4$ $U$ |
| $5$ $E$ |
