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STD 11 - 6. system of particles and rotational motion — NEET STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceNEETSTD 11 - 6. system of particles and rotational motion4 Marks
Question
$A$ solid cone hangs from $a$ frictionless pivot at the origin $O$, as shown. 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 $\tau_Z$ 
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Answer

Torque $T=\vec{r} \times \vec{F}$

For $F=a \hat{k}$ where $P$ is $(0, b,-c)$

$T=(b \hat{j}-c \hat{k}) \times(a \hat{k})=a b \hat{i}$

For $F=-a \hat{k}$ where $P$ is $(0,-b,-c)$

$T=(-b \hat{j}-c \hat{k}) \times(-a \hat{k})=a c \hat{i}$

For $F=a \hat{j}$ where $P$ is $(-b, 0,-c)$

$T=(-b \hat{i}-c \hat{k}) \times(a \hat{j})=-a b \hat{k}+a c \hat{i}$

Therefore it has negative $z$ component.

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