Rotation Around A Fixed Axis - Vector Expression

Vector Expression

The above development is a special case of general rotational motion. In the general case, angular displacement, angular velocity, angular acceleration and torque are considered to be vectors.

An angular displacement is considered to be a vector, pointing along the axis, of magnitude equal to that of . A right-hand rule is used to find which way it points along the axis; if the fingers of the right hand are curled to point in the way that the object has rotated, then the thumb of the right hand points in the direction of the vector.

The angular velocity vector also points along the axis of rotation in the same way as the angular displacements it causes. If a disk spins counterclockwise as seen from above, its angular velocity vector points upwards. Similarly, the angular acceleration vector points along the axis of rotation in the same direction that the angular velocity would point if the angular acceleration were maintained for a long time.

The torque vector points along the axis around which the torque tends to cause rotation. To maintain rotation around a fixed axis, the total torque vector has to be along the axis, so that it only changes the magnitude and not the direction of the angular velocity vector. In the case of a hinge, only the component of the torque vector along the axis has effect on the rotation, other forces and torques are compensated by the structure.