Physics

Torque

[latexpage]

Note: for the purposes of this article, $ \bold{ x } $ and \overrightarrow{x} are equivalent notations for a column vector

Introduction

Figures 1 and 2 depict a rod-shaped object accelerating due to a force \overrightarrow{F} applied at its center of mass m_c

center of mass rod
Figure 1
torque2
Figure 2
  • For the cases depicted in Figures 1 and 2,
    • The rod is (linearly) accelerating (represented by pink arrows) in the same direction as \overrightarrow{F}
    • We can use \overrightarrow{F} = m \overrightarrow{a} to find the acceleration of the object:

\overrightarrow{a}_{\mathrm{object}} = \overrightarrow{F} / m_c

Question: what happens when a force is applied to an object at a region that is not at the center of mass?

  • Answer: assuming the object is free-floating in 3D space, it will rotate about the center of mass (see figure 3)
    torque3
    Figure 3: rotation about the center of mass in free 3D space
    • Notes:
      • \overrightarrow{F}_\perp denotes the component of the force perpendicular to the object with respect to its center of mass
      • Unlike cases 1 and 2, the resulting movement of the object (pink arrows) in case 3 is not in the same direction as \overrightarrow{F}_\perp

Now consider an object similar a hand on a clock (figures 4 and 5) where one component of this object is fixed in space (called the axis of rotation, pivot point, or fulcrum) and two other components (the arms) comprise a single rigid rod that is free to rotate about the pivot point (blue dot):

torque5 - Copy
Figure 4: rigid rotor with zero torque
torque_rigidrotor1
Figure 5: rigid rotor with positive torque
  • Note: \overrightarrow{r} denotes the moment arm distance or a vector whose origin starts at the pivot point and ends at the point of impact by a force \overrightarrow{F}_\perp who direction is perpendicular to \overrightarrow{r}
  • Applying a force at the pivot point wont cause the object to move (see case 4):
  • BUT if we apply a force on a “clock-hand” component, it will accelerate in the direction that causes it to rotate about the pivot point (see case 5):

Definitions

Torque (units in \mathrm{Nm}) describes the effect of \overrightarrow{F}_\perp on an object with respect to the moment arm distance \overrightarrow{r} at which a force is applied

  • Equation:
    • Let F_\perp = \left| \overrightarrow{F}_\perp \right| and r = \left| \overrightarrow{r} \right|
    • \tau =\left| \overrightarrow{\tau} \right| = \overrightarrow{r} \times \overrightarrow{F}_\perp = r F_\perp
  • Note: sometimes, people use the term “moment” to describe torque
  • Importantly, even though torque has units in \mathrm{Nm} (same SI units for work ⇒ 1 \mathrm{J} = 1 \mathrm{Nm}), we DON’T assign it units in joules
    • Work ⇒ the subsequent change in an object’s position when a force applied is translational (i.e., non-rotating)
      • Vectors describing  the force and object acceleration have the same direction (i.e. they are parallel)
      • e.g., cases 1 and 2
    • Torque ⇒ the subsequent change in an object’s position with respect to a pivot point when a force is applied  is rotational
      • Vectors describing the force and moment arm distance are perpendicular relative to each other
      • Vectors describing the force and object acceleration have different directions
      • e.g., cases 3, 4, and 5

Torque is also be defined as the rate of change of angular momentum of an object

Torque convention:

  • Positive torque (\tau > 0) ⇒ counterclockwise rotation
  • Negative torque (\tau < 0) ⇒ clockwise rotation

Sources

 

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