In the case of circular motion, the linear tangential speed of a particle at a radius from the axis of rotation is related to the angular velocity by the relation:

Sigma notation, also known as summation notation, provides a concise method for representing the sum of a sequence of terms that follow a regular pattern. It utilizes the uppercase Greek letter sigma (∑), A typical expression is:In this form, k the index of summation is 1, the starting value, and n the ending value. The term ak​ represents the general term of the sequence.For example, the increasing sequence 5, 7, 9, ..., 23 over 10 terms can be expressed as:This simplifies the representation...

Altruistic behaviors are “unselfish” behaviors—those that help another individual at the expense of the individual carrying out the behavior. Despite the negative consequences for the altruistic animal, these behaviors are thought to have evolved for several reasons.

A two-dimensional system in mechanical engineering involves the analysis of motion and forces in a plane. A two-dimensional force vector can be resolved into its components as:

Voluntary behavior with the intent to help other people is called prosocial behavior. Why do people help other people? Is personal benefit such as feeling good about oneself the only reason people help one another?

The torque-free motion refers to the movement of a rigid body in space when no external torques are acting upon it. This type of motion can be observed in environments where there are no external forces or frictions, like in outer space. For example, a rotation of Mars in space is a torque-free motion. Mars is an axisymmetric object, meaning it has an axis of symmetry along which it rotates, designated as the z-axis. The rotating frame of reference is defined such that the center of mass of...