Study Notes on Torque

Torque

• Definition: The turning effect of a force produced on a body about an axis is called torque.

• Mathematical Formulation:

  • Torque can be defined as the product of the magnitude of force and the perpendicular distance from the axis of rotation to the line of action of the force.
  • Torque ( \tau ) can also be represented as a vector product of the radius ( \mathbf{r} ) from the axis of rotation to the point of application of the force and the force vector ( \mathbf{F} ).
  • Mathematically, torque can be expressed as:
    \tau = \mathbf{r} \times \mathbf{F}
  • Alternatively, in terms of distance and force, torque can be represented as: \tau = L \cdot F
    • Where:
    • ( L ) = Perpendicular distance (moment arm) from the axis of rotation to the line of action of the force.
    • ( F ) = Magnitude of the applied force.

• Dependables of Torque:

  • Torque depends on the following factors:
  1. Magnitude of Force: Greater force applied increases torque.
  2. Perpendicular Distance: The distance from the axis of rotation to the line of action of the force affects the torque (known as the moment arm).

• Units of Torque:

  • The SI unit of torque is Newton's meter (N·m).
  • Dimensions of torque: ( [ML^2T^{-2}] )
  • Where:
    • ( M ) = Mass unit (kg)
    • ( L ) = Length unit (m)
    • ( T ) = Time unit (s)

• Examples of Torque:

  • Tightening a Knot: When using a spanner or wrench, the user applies force away from the axis of rotation, creating torque to tighten the knot.
  • Seesaw: A seesaw rotates on and off the ground due to torque imbalance when weights are placed on opposite sides.

• Torque in Rigid Bodies:

  • Consider a rigid body where:
  • ( F ) is the force acting on the body at point B.
  • ( \mathbf{R} ) is the position vector of point B with respect to the pivot point O.
  • ( \theta ) is the angle between vector ( \mathbf{r} ) and radius vector ( \mathbf{R} ).