A particle of mass m is moving along the side of a square of side a
By Problem: A particle of mass m is moving along the side of a square of side a, with a uniform speed v in the x-y plane as shown in the figure. Which of the following statements is false for the angular momentum $\vec{L}$ about the origin? (JEE Mains 2016)
- $\vec{L}=-\frac{mv}{\sqrt{2}} R\,\hat{k}$ when the particle is moving from A to B.
- $\vec{L}=mv\left[\frac{R}{\sqrt{2}}-a\right]\hat{k}$ when the particle is moving from C to D.
- $\vec{L}=mv\left[\frac{R}{\sqrt{2}}+a\right]\hat{k}$ when the particle is moving from B to C.
- $\vec{L}=\frac{mv}{\sqrt{2}} R\,\hat{k}$ when the particle is moving from D to A.
Solution: The angular momentum of a particle of mass $m$ moving with a velocity $\vec{v}$ at the position $\vec{r}$ is given by \begin{align} \vec{L}=m\vec{r}\times\vec{v}=mvr_\perp \, \hat{n}, \nonumber \end{align} where $r_\perp$ is the component of $\vec{r}$ perpendicular to $\vec{v}$ (perpendicular distance or lever arm) and $\hat{n}$ is the unit vector in the direction of $\vec{r}\times\vec{v}$. Note that angular momentum is about the origin as $\vec{r}$ is a position vector from the origin to the current location of the particle. These quantities on given paths of the particle are \begin{align} &\mathrm{AB:} && r_\perp=\tfrac{R}{\sqrt{2}}, &&\vec{L}=mv\left(\tfrac{R}{\sqrt{2}}\right)(-\hat{k}),\nonumber\\ &\mathrm{BC:} && r_\perp=\tfrac{R}{\sqrt{2}}+a, &&\vec{L}=mv\left(\tfrac{R}{\sqrt{2}}+a\right)\hat{k},\nonumber \\ &\mathrm{CD:} && r_\perp=\tfrac{R}{\sqrt{2}}+a, &&\vec{L}=mv\left(\tfrac{R}{\sqrt{2}}+a\right)\hat{k},\nonumber\\ &\mathrm{DA:} && r_\perp=\tfrac{R}{\sqrt{2}}, &&\vec{L}=mv\left(\tfrac{R}{\sqrt{2}}\right)(-\hat{k}),\nonumber \end{align} Note that $\hat{n}$ is into the paper on the paths AB and DA.
More Solved Problems on Angular Momentum from IIT JEE
- A small mass m is attached to a mass string (IIT JEE 2012)
- A particle of mass 2 kg is on a smooth horizontal table (JEE Mains 2015)
- A particle of mass 20 g is released with an initial velocity 5 m/s along the curve (JEE Mains 2019)
- The time dependence of the position vector of a particle (JEE Mains 2019)
- Angular Momentum of a Projectile
- Angular Momentum
