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{\bf Question}

\begin{description}
\item[(a)] Can the torque on a particle be zero without the force
being zero?  Explain.
\item[(b)] Can the force on a particle be zero without the
angular momentum being zero? Explain.
\end{description}

\vspace{.25in}

{\bf Answer}

\begin{description}
\item[(a)]
If the torque is zero then ${\bf r} \times {\bf F} = 0$

Now ${\bf N} = {\bf r} \times {\bf F}$ hence ${\bf N} = 0
\Rightarrow$ {\bf r} or {\bf F} is zero, or {\bf r} and {\bf F}
are parallel.

i.e. yes.  either ${\bf r} = 0$ (the force is acting at the
origin) or {\bf r} and {\bf F} are parallel; and so {\bf F} is
acting in a direction through the origin.

\item[(b)]
${\bf F} = 0 \Rightarrow {\bf N} = {\bf r} \times {\bf F} = 0$

Using Newton's 2nd law: $ 0 = {\bf N} = \dot {\bf L} \Rightarrow
{\bf L} = $ constant. Thus {\bf L} just needs to be constant (not
necessarily zero).

Answer: Yes.

\end{description}

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