When a force $\backslash (\backslash$mathbf$\{$F$\}\backslash )$ acts on an object at some distance $\backslash ($ r $\backslash )$ from a point $\backslash ($ P $\backslash ).$ called the pivot point, it can cause the object to rotate rather than translate. The ability of a force to cause rotation is called the torque about $\backslash ($ P $\backslash )$ due to the force F and is symbolized by the Greek lowercase letter tau $\backslash (\backslash$tau $\backslash ).$ Torque equals the cross product between the applied force $\backslash (\backslash$mathbf$\{$F$\}\backslash )$ and the distance vector $\backslash (\backslash$mathbf$\{$r$\}=\backslash$overrightarrow$\{$P Q$\}\backslash )$ that points from $\backslash ($ P $\backslash )$ to the point $\backslash ($ Q $\backslash )$ where $\backslash (\backslash$mathbf$\{$F$\}\backslash )$ acts on the object. That is$,$

$\backslash [$

$\backslash$tau$=\backslash$mathbf$\{$r$\}\backslash$times $\backslash$mathbf$\{$F$\}$

$\backslash ]$

The figure shows a wrench that is being used to turn a nut and three separate forces that could be applied to the end of the wrench. The point $\backslash ($ P $\backslash )$ is at the origin of the coordinate system and the point $\backslash ($ Q $\backslash )$ is at the end of the wrench at $\backslash ((0\mathrm{.}25,0,0)\backslash ),$ where the forces are applied.

$($a$)$ Describe the direction of the positive $\backslash ($ z $\backslash )-$axis, given the axes shown in the figure. Assume that it is a right$-$handed system.

along the $\backslash (\backslash$mathbf$\{$j$\}\backslash )-$axis

out of the ij$-$plane

along the $\backslash ($ i $\backslash )-$axis

parallel to the ij$-$plane

in the direction of $\backslash (\backslash$mathbf$\{$F$\}\_\{2\}\backslash )$

into the ij$-$plane

Correct Answer

$($b$)$ Find the torque $\backslash ($ r $\backslash )$ about $\backslash ($ P $\backslash )$ due to each of the three forces shown in the figure. Assume that the magnitude of each force is $230$ N and the wrench is $0\mathrm{.}25$ m long.

$($Use symbolic notation and fractions where needed. Let $\backslash (\backslash$boldsymbol$\{$t$\}\_\{$i$\}\backslash )$ be the torque due to $\backslash (\backslash$mathbf$\{$F$\}\_\{$t$\}\backslash ).)$

$\backslash [$

$\backslash$boldsymbol$\{$r$\}\_\{1\}=$

$\backslash ]$

Incorrect Answer

$\backslash [$

$\backslash$tau$\_\{2\}=81\backslash$mathrm$\{$k$\}$

$\backslash ]$

incorrect Answer