Solution
= Solution
Let an upward <force> $P$ act along $x=0$ and a downward parallel <force> $Q$ along $x=d>0$. An equivalent <resultant force> must preserve both total <force> and <torque>. Its signed vertical component is $R=P-Q$, and taking <torques> about the first line gives $Rx_R=-Qd$. Thus
$$
\boxed{R=P-Q,\qquad x_R=-\frac{Qd}{P-Q}\quad(P\ne Q).}
$$
Its magnitude is $|P-Q|$ and its direction agrees with the larger <force>. Its line lies outside the two original lines on the side of that larger <force>; for example, $P>Q$ gives $x_R<0$. The products of the original <forces> and their perpendicular distances from this line are equal.
If $P=Q\ne0$ and $d\ne0$, the total <force> vanishes but the <torque> is $-Pd$. \b[No single force is equivalent to this system:] it is a <force couple> of magnitude $Pd$. Its <torque> is independent of the reference point because translating that point changes the <torque> by the <cross product> of the shift and the zero <resultant force>. Equal opposite <forces> on the same line instead cancel completely.
\Image[/past-exam-of-the-medicine-course-of-the-university-of-oxford/1867/first-examination/physics-1867-opposed-forces.png]
{title=Unequal opposed forces have an external resultant line; equal separated forces form a couple}
The <eye> is rotated by six <extraocular muscles>. The rectus <muscles> approach it from behind; the oblique <muscles> have oblique lines of pull, with the <superior oblique muscle> tendon redirected by an orbital pulley. Attachments on the <sclera> give <lever arms> about the rotation centre. In the straight-ahead position their actions are as follows; vertical and torsional contributions vary with gaze direction.<Neuroscience, 2nd edition edited by D. Purves and others>[Chapter 20, “The Actions and Innervation of Extraocular Muscles”: antagonistic pairs, axes and eye-position dependence]{ref}
|| <Extraocular muscle>
|| Actions
|| Innervation
| <Medial rectus muscle>
| Turns the <eye> inward
| <Oculomotor nerve>, III
| <Lateral rectus muscle>
| Turns the <eye> outward
| <Abducens nerve>, VI
| <Superior rectus muscle>
| Elevates; also turns inward and rotates the upper pole inward
| <Oculomotor nerve>, III
| <Inferior rectus muscle>
| Depresses; also turns inward and rotates the upper pole outward
| <Oculomotor nerve>, III
| <Superior oblique muscle>
| Rotates the upper pole inward; also depresses and turns outward
| <Trochlear nerve>, IV
| <Inferior oblique muscle>
| Rotates the upper pole outward; also elevates and turns outward
| <Oculomotor nerve>, III
Coordinated contraction selects the desired rotation and balances unwanted components. This illustrates <torque> control; the full ocular geometry is not simply the parallel-force pair of the first calculation.