Rotational Analogue of a Force: Moment of a Force Or Torque

Torque is the rotational analogue of force, which means it is the quantity that causes angular motion in objects. Just like a force causes linear motion, torque causes rotational motion. Torque depends not only on how much force we apply but also on where we apply it and the angle at which we apply it. For example, when opening a door, we push away from the hinges to make opening easier because the torque is greatest at a greater distance from the axis. Understanding torque helps us understand why some tasks become easier when we apply forces at certain points and in specific directions.

The rotational analogue of a force, defined as the quantity that represents the rotational ability of a force. It depends upon the magnitude of the force, the point of application of the force, and the angle between the direction of the force and the line joining the axis of rotation with the point of application.

Vector Form

\[\vec τ\] = \[\vec r\] × \[\vec F\]

Here:

• \[\vec τ\] = Torque
• \[\vec r\] = Position vector of the point of application of force from the axis of rotation
• \[\vec F\] = applied force

Scalar (Magnitude) Form

τ = r F sin θ

Where:

• r =  Perpendicular distance from the axis of rotation
• F = Magnitude of the force
• θ = The smaller angle between the direcions of \[\vec r\] and \[\vec F\]

• Magnitude of Force: The Greater the force applied, the greater the torque. For heavier objects (larger mass), we need a proportionally larger force to produce the same angular displacement.
• Distance from Axis of Rotation: The Greater the perpendicular distance of the point of application from the axis of rotation, the greater the torque. The force should be applied as far away as possible from the axis.
• Angle of Application: The angle between the direction of the applied force and the position vector affects torque. Maximum torque occurs when this angle is 90° (force applied perpendicular to the position vector).

The direction of torque is determined using the right-hand thumb rule or the cross product rule.

Right-Hand Thumb Rule: Point your fingers in the direction of \[\vec r\] (position vector), curl them towards \[\vec F\] (force direction), and your thumb points in the direction of torque \[\vec τ\].

Special Convention for Representing Direction:

• ☉ (Circle with dot): Torque is perpendicular to the plane of paper and coming out (towards you)

• ⊗ (Circle with cross): Torque is perpendicular to the plane of the paper and going in (away from you)

The direction of torque is always perpendicular to the plane containing both \[\vec r\] and \[\vec F\].

• When force or distance increases: If r or F is greater, the torque and hence the rotational effect is greater. This is why we apply force away from the hinges when opening a door.
• When the angle is 90°: If θ = 90°, then τ = τ_max = rF. Maximum torque is produced. The force should always be applied along the normal (perpendicular) direction for easy rotation.
• When angle is 0° or 180°: If θ = 0° or θ = 180°, then τ = τ_min = 0. If the force is applied parallel or anti-parallel to \[\vec r\], there is no rotation at all.
• Dependence on point of application: Torque depends not only on the magnitude and direction of force but also on the point where the force acts with respect to the axis of rotation. The same force can produce different torques depending on where it is applied.

• Opening a Door
  Pushing far from the hinges and at 90° gives maximum torque—easier to open.
• Steering Wheel
  Holding the rim (far from the center) and turning gives more torque with less force.
• Using a Spanner
  A longer spanner gives more torque than a shorter one with the same force.
• Seesaw or Lever
  A lighter person far from the pivot can balance a heavier person close to it.
• Bicycle Pedaling
  Pedaling when the pedal is farthest from the axle gives better torque.
• Loosening a Nut
  A longer wrench makes loosening easier by increasing torque.