Question

A solid cone hangs from a frictionless pivot at the origin O as shown in the figure. If $\hat{i},\hat{j}$ and $\hat{k}$ are unit vectors, and a, b, and c are positive constants, which of the following forces $F$ applied to the rim of the cone at a point P, results in a torque $\tau$ on the cone with a negative component ${\tau }_z$?

• A. $F=a\hat{k}$, P is $(0,b,-c)$
• B. $F=-a\hat{k}$, P is $(0,-b,-c)$
• C. $F=a\hat{j}$, P is $(-b,0,-c)$
• D. $None$

Answer 1

The correct option is C $F=a\hat{j}$, P is $(-b,0,-c)$

Torque $T=$ $\overrightarrow{r}\times \overrightarrow{F}$
For $F=a\hat{k}$ where P is $(0,b,-c)$
$T=(b\hat{j}-c\hat{k})\times \left(a\hat{k}\right)=ab\hat{i}$
For $F=-a\hat{k}$ where P is $(0,-b,-c)$
$T=(-b\hat{j}-c\hat{k})\times (-a\hat{k})=ac\hat{i}$
For $F=a\hat{j}$ where P is $(-b,0,-c)$
$T=(-b\hat{i}-c\hat{k})\times \left(a\hat{j}\right)=-ab\hat{k}+ac\hat{i}$
Therefore it has negative $z$ component.