A particle of mass 2 kg is on a smooth horizontal table
By Problem: A particle of mass 2 kg is on a smooth horizontal table and moves in a circular path of radius 0.6 m. The height of the table from the ground is 0.8 m. If the angular speed of the particle is 12 rad/second, the magnitude of its angular momentum about a point on the ground right under the centre of the circle is (JEE Mains 2015)
- 8.64 kg-m2/s
- 11.52 kg-m2/s
- 14.4 kg-m2/s
- 20.16 kg-m2/s
Solution: The particle of mass m = 2 kg moves on a horizontal circle of radius $r_0=0.6$ m centered at the point C (see figure). The point C is at a height $h=0.6$ m from the ground point O.
The particle moves on base circle of a cone of radius $r$, height $h$ and vertex O. If the particle is at the point P then its position vector $\vec{r}$ lies on the curved surface of the cone (it is generator of the cone). The length of the position vector is \begin{align} |\vec{r}|=\sqrt{r_0^2+h^2}. \end{align} The velocity $\vec{v}$ of the particle is tangential to the base circle. The magnitude of velocity is \begin{align} |\vec{v}|=\omega r_0, \end{align} where $\omega=12$ rad/second is angular speed of the particle in its circular path.
It is not difficult to see that $\vec{r}$ and $\vec{v}$ are perpendicular to each other (we encourage you to try it on a paper cone and get yourself convinced). Thus, magnitude of the angular momentum of the particle about the point O is \begin{align} |\vec{L}|&=m|\vec{r}\times\vec{v}| \\ &=m|\vec{r}|\,|\vec{v}|\sin90^\circ\nonumber\\ &=mwr_0\sqrt{r_0^2+h^2} \\ &=14.4\;\mathrm{kg\,m^2/s}.\nonumber \end{align} We encourage you to write the expressions for $\vec{r}$ and $\vec{v}$ and then get $\vec{L}$ by cross product. You will find that $z$-component of $\vec{L}$ is independent of $h$ i.e., it is independent of the choice of the origin O. However, component in $x\text{-}y$ plane depends on $h$. Can you see that $\vec{L}$ moves on a cone as the particle moves in the circle?
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 m is moving along the side of a square (JEE Mains 2016)
- 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
