Physics · Ch 4 — Laws of Motion
Rotational Analogue of a Force -- Moment of a Force or Torque
Rotational Analogue of a Force -- Moment of a Force or Torque
When opening a door, applying the force far from the hinge (and roughly perpendicular to the door) makes it easy to open; applying the same force close to the hinge, or nearly parallel to the door, makes it very hard to produce the same rotation -- and a heavier door needs proportionally more force for the same effect. So the ROTATIONAL ABILITY of a force depends on three things: the mass of the object being rotated, the perpendicular DISTANCE of the point of application of the force from the axis of rotation (further is more effective), and the ANGLE between the force and the line joining the axis to the point of application (most effective at 90 degrees).
Combining these, the quantity TORQUE (or moment of a force) is defined as the rotational analogue of force: , where is the applied force and is the position vector of its point of application, measured from the axis of rotation. Since rotation has a definite sense (clockwise/anticlockwise), torque must be a VECTOR quantity, with its direction given by the right-hand rule (perpendicular to the plane containing and ) -- conventionally drawn using a dot-in-circle symbol for a vector coming out of the page (towards the viewer) and a cross-in-circle symbol for a vector going into the page (away from the viewer), based on the visual mnemonic of an arrow's tip (seen approaching) versus its tail feathers (seen departing). …
What this figure shows. Part (a) is a 3-D perspective drawing of a flat, laminar (plane) object free to rotate about a fixed axis AOB that passes through the object perpendicular to its plane; a force F is applied at a point away from the axis, with its position vector r drawn from a point on the axis to the point of application. Parts (b) and (c) are TOP VIEWS of the same object looking down along the axis: in (b) the force F and position vector r are drawn in the plane of the page such that the resulting rotation is ANTICLOCKWISE, with the torque vector (by the right-hand rule) pointing straight OUT of the page towards the viewer; in (c) the force and r are arranged so the rotation is CLOCKWISE instead, with the torque vector pointing straight INTO the page, away from the viewer. Both (b) and (c) show the …
What this figure shows. A four-part legend explaining the dot/cross convention for vectors not lying in the plane of the figure. Part (a) shows a traditional arrow (shaft with a triangular head and trailing feathers, like an actual archery arrow) used as the mnemonic. Part (b) shows a circle with a single dot at its centre, representing the arrow's TIP as seen coming towards the viewer -- i.e., a vector pointing OUT of the page, towards the reader. Part (c) shows a circle with a cross (X or two crossed lines) inside it, representing the arrow's tail FEATHERS as seen from behind as the arrow flies away -- i.e., a vector pointing INTO the page, away from the reader. Part (d) shows a straight line with both a dot-circle and a cross-circle marked along it at different points, illustrating a single line of action that is perpendicular to the plane of the figure, combining both conv …