Introduction to Rotation, angular velocity/acceleration and moments

An object rotates when it moves in a circular motion around a rotation axis. An infinite number of rotation axes can exist for a three-dimensional object. The body is said to be auto rotating or spinning if the rotation axis passes internally through its own centre of mass, and the axis’s surface intersection is called a pole. Angular velocity, often known as rotational velocity, is a pseudovector description of how quickly the angular position or orientation of an object varies with time in physics. The time rate of change of angular velocity is known as angular acceleration in physics. Because there are two types of angular velocity, spin angular velocity and orbital angular velocity, there are two types of angular acceleration as well.

Rotation

A rotation, unlike a translation, is a stiff body movement that keeps a point fixed. This definition applies to both two and three-dimensional rotations (in a plane and in space, respectively.) Rotations, translations, and combinations of the two constitute all rigid body movements. A rotation is just a radial rotation that leads to a common point. The axis of such motion contains that common point. The axis is perpendicular to the motion plane at 90 degrees. A third rotation occurs when a rotation around a point or axis is followed by another revolution around the same point or axis. A rotation’s inverse (opposite) is also a rotation. As a result, the rotations around a point/axis form a collection. A rotation around one point or axis and a rotation around another point or axis, on the other hand, may result in something other than a rotation, such as a translation.

Axis of rotation

A specific case of rotational motion is rotation around a fixed axis. The fixed-axis theory rules out the notion of an axis changing its orientation, hence wobbling and precession aren’t possible. Euler’s rotation theorem states that simultaneous rotation along multiple stationary axes is impossible; if two rotations are forced at the same time, a new axis of rotation will appear. Because it is an imagined entity, understanding the axis of rotation can be challenging. The axis of rotation is an imaginary line that passes through the pivot/rotation point of a joint in human anatomy (for example, the axis of rotation for flexing and extending the arm projects through the elbow joint). Each of the three planes of motion (sagittal, frontal, and transverse) is rotated around an axis.

Angular velocity

The rotation rate of an object is measured in angular velocity, which is a vector measure of how quickly it rotates or revolves relative to another point. Angular velocity is the pace at which an item rotates or circles around an axis in time. The Greek letter omega , occasionally) represents angular velocity. The SI unit of angular velocity is radians per second because it is measured in angle per unit time.

Formula of angular velocity 

Angular velocity ω=dθdt

Here dθ is the  rate of change of position

And dt is the rate of change of time.

Angular acceleration

The angular acceleration is the rate at which the angular velocity changes over time, and it’s commonly given in radians per second per second.

Formula of angular acceleration

Angular acceleration α= dωdt

Here dωdt is the rate of change of angular velocity

vector equation for moment

The vector or cross product approach is an alternative to computing the moment using scalar values. Using scalar values is usually easier for simple two-dimensional situations, but the cross product method is usually easier for more complex problems. The torque about a point is equal to the cross product of a vector r from the point to anywhere on the force’s line of action and the force vector itself, according to the cross product method.

 τ= r ×F

Conclusion

Mathematics and physics are both concerned with rotation. It is based on the rotation or movement of objects around the axis’s centre. The earth rotates around its own axis and revolves around the sun in real life. The formulas for rotation and the degree of rotation are used to determine rotation. The temporal rate of change of angular velocity, commonly indicated by and expressed in radians per second, is referred to as angular acceleration. Furthermore, because the angular acceleration varies linearly with time, it is constant and does not depend on the time variable