Chapter 11 Rotational Dynamics and Torque

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11.0 Torque

Torque is defined as the quantity that measures the tendency of a force to rotate a body about some axis. It is the rotational analogue of force or simply rotational force. That is, torque plays the role of force in rotational motion. The equation for torque is force times the distance.

The equation for torque is: (rotational force, torque, units: Nm) (11.1)

where d is the distance perpendicular to the axis of rotation, and F is the tangential force acting on the object.

Fig. 11.1 Rotational action.

Since torque is a vector, it can be calculated from the cross product of the radius vector, , and the force vector, , with axis of rotational at point 0 (see Fig. 11.1), which can write mathematically as:

(torque, units: Nm) (11.2)

where F is force and r is the distance in the lever arm. Its magnitude is given by

(11.3)

Equation (11.3) sates that torque can also be defined as the force perpendicular to the radius (r) times the distance from the axis (Sometimes "r" is also called the moment arm of F) multiplied by the sine of angle between them, see Fig. 11.2.

Text Box:

Fig. 11.2 A wheelbarrow.

Equation (11.3) can also be written as:

(11.4)

where is the displacement distance or the lever arm. This shows that the torque depends on the angle between the applied force and the radius or the lever arm. A wheelbarrow is a very good example of application of torque in doing work, see Fig 11.2.

Text Box:

Fig. 11.3 A hinged door.

Another good example of torque or rotational force is exhibited by opening or closing a door. Applying a force to a door produces an angular acceleration that varies with the point at which the force is applied, with the edge of the door exhibiting greater force and higher angular acceleration.

Analysis of closing and opening a door - A door rotates on a hinge when opened and closed. If a force of 10 N is applied to the door, it will swing open. If you apply the force at point A, the door will swing open very fast. If you apply the force at point B, the door will still swing open, just at a slower rate. Torque depends on the magnitude and where the force acts in respect to the axis.

If you apply the force on the edge of the door towards the hinges, there will be no rotational acceleration. There will be a lever arm equal to zero, which means there is zero torque.

� The lever arm is the perpendicular distance from the axis of rotation to the line along which the force acts.

� Torque relies on the force of the magnitude, the orientation of the force on the object, and the angle of the force in respect to the axis.

� Torque is a vector in the sense that its direction is along the axis

� I, or the momentum of inertia, depends on the mass (how big) of the object and on the distribution of the mass in respect to the axis

Text Box: Fig. 11.4a