Obtain relation between linear and angular …

Relation between linear velocity and angular velocity

Let us consider the randomly shaped body undergoing a rotational motion as shown in the figure below. The linear velocity of the particle is related to the angular velocity. While considering the rotational motion of a rigid body on a fixed axis, the extended body is considered as a system of particles moving in a circle lying on a plane that is perpendicular to the axis, such as the center of rotation lies on the axis.

In this figure, the particle P has been shown to rotate over a fixed axis passing through O. Here, the particle represents a circle on the axis. The radius of the circle is the perpendicular distance between point P and the axis. The angle indicates the angular displacement Δθ of the given particle at time Δt. The average angular velocity in the time Δt is Δθ/Δt. Since Δt tends to zero, the ratio Δθ/Δt reaches a limit which is known as the instantaneous angular velocity dθ/dt. The instantaneous angular velocity is denoted by ω.

From the knowledge of circular motion, we can say that the magnitude of the linear velocity of a particle traveling in a circle relates to the angular velocity of the particle ω by the relation υ/ω= r, where r denotes the radius. At any instant, the relation v/ r = ω applies to every particle that has a rigid body.

If the perpendicular distance of a particle from a fixed axis is r_i, the linear velocity at a given instant v is given by the relation,

V_i = ωr_i

Similarly, we can write the expression for the linear velocity for n different particles comprising the system. From the expression, we can say that for particles lying on the axis, the tangential velocity is zero as the radius is zero. Also, the angular velocity ω is a vector quantity which is constant for all the particles comprising the motion.