Question

A conical pendulum consists of a mass $M$ suspended from a strong sling of length $l$. The mass executes a circle of radius $R$ in a horizontal plane with speed $v$. At time $t$, the mass is at position $R\hat{i}$ and has $v\hat{j}$ velocity. At time $t$, the angular momentum vector of mass $M$ about the point from which the string passes on the ceiling is

• A. $MvR\hat{k}$
• B. $Mvl\hat{k}$
• C. $Mvl[\sqrt{\frac{l^2-R^2}{l^2}}\hat{i}+\frac{R}{l}\hat{k}]$
• D. $-Mvl[\sqrt{\frac{l^2-R^2}{l}}\hat{i}+\frac{R}{l}\hat{k}]$

Answer 1

Answer: (C)

$Mvl[\sqrt{\frac{l^2-R^2}{l^2}}\hat{i}+\frac{R}{l}\hat{k}]$

Angular momentum about O $\overrightarrow{L_O}=M(\overrightarrow{OM}\times \overrightarrow{v})$

$\overrightarrow{L_O}=M\left(\right[\overrightarrow{OR}+\overrightarrow{RM}]\times \overrightarrow{v})$

$\overrightarrow{L_O}=M\left(\right[-\sqrt{l^2-R^2}\hat{k}+R\hat{i}]\times v\hat{j})$

$\overrightarrow{L_O}=M[v\sqrt{l^2-R^2}\hat{i}+Rv\hat{k}]$

$\overrightarrow{L_O}=Mvl[\frac{\sqrt{l^2-R^2}}{l}\hat{i}+\frac{R}{l}\hat{k}]$