System of Particles and Rotational Dynamics

Angular Velocity and Angular Acceleration

Angular velocity is the rate of velocity at which an object or a particle is rotating around a center or a specific point in a given time period. It is also known as rotational velocity. Angular velocity is measured in angle per unit time or radians per second (rad/s). The rate of change of angular velocity is angular acceleration. Let us learn in more detail about the relation between angular velocity and linear velocity, angular displacement and angular acceleration.

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Angular Velocity

Angular velocity plays an eminent role in the rotational motion of an object. We already know that in an object showing rotational motion all the particles move in a circle. The linear velocity of every participating particle is directly related to the angular velocity of the whole object.

These two end up as vector products relative to each other. Basically, the angular velocity is a vector quantity and is the rotational speed of an object. The angular displacement of in a given period of time gives the angular velocity of that object.

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Relation Between Angular Velocity and Linear Velocity

For understanding the relation between the two, we need to consider the following figure:

relation between angular velocity and linear velocity

The figure above shows a particle with its center of the axis at C moving at a distance perpendicular to the axis with radius r. v is the linear velocity of the particle at point P. The point P lies on the tangent of the circular motion of the particle. Now, after some time(Δt)  the particle from P displaces to point P1. Δθ or ∠PCP1 is the angular displacement of the particle after the time interval Δt. The average angular velocity of the particle from point P to P1 = Angular displacement / Time Interval = Δθ/Δt

At smallest time interval of displacement, for example, when Δt=0 the rotational velocity can be called an instantaneous angular (ω) velocity, denoted as dt/dθ for the particle at position P. Hence, we have ω = dt/dθ

Linear velocity (v) here is related to the rotational velocity (ω ) with the simple relation, v= ωr, r here is the radius of the circle in which the particle is moving.

Angular Velocity of a Rigid Body

This relation of linear velocity and angular velocity apply on the whole system of particles in a rigid body. Therefore for any number of particles; linear velocity vi = ωri

‘i’ applies for any number of particles from 1 to n. For particles away from the axis linear velocity is ωr while as we analyze the velocity of particles near the axis, we notice that the value of linear velocity decreases. At the axis since r=0 linear velocity also becomes a zero. This shows that the particles at the axis are stationary.

A point worth noting in case of rotational velocity is that the direction of vector ω does not change with time in case of rotation about a fixed axis. Its magnitude may increase or decrease. But in case of a general rotational motion, both the direction and the magnitude of angular velocity (ω) might change with every passing second.

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Rotational Velocity of Revolutions

When a rigid body rotates around an axis, after the lapse of some time it completes a revolution. The time taken by that rigid body to complete a revolution is called the frequency of that body. Rotational velocity and frequency hence have a relation between each other. Here one revolution is equal to 2π, hence ω= 2π/ T

The time taken to complete one revolution is T and ω= 2π f. ‘f’ is the frequency of one revolution and is measured in Hertz.

Angular Acceleration

When an object follows a rotational path, it is said to move in an angular motion or the commonly known rotational motion. In the course of such motion, the velocity of the object is always changing. Velocity being a vector involves a movement of an object with speed that has direction. Now, since in a rotational motion, the particles tend to follow a circular path their direction at every point changes constantly. This change results in a change in velocity. This change in velocity with time gives us the acceleration of that object.

Angular acceleration is a non-constant velocity and is similar to linear acceleration of translational motion. Understanding linear displacement, velocity, and acceleration are easy and this is why when we intend to study rotational motion, we compare its vectors with translational motion. Like linear acceleration, angular acceleration (α) is the rate of change of angular velocity with time. Therefore, α = dω/ dt

Now since for rotation about a fixed axis the direction of angular velocity is fixed therefore the direction of angular momentum α is also fixed.  For such cases, the vector equation transforms into a scalar equation.

Solved Question For You

 1. The angular velocity of a scooter tire of diameter 6 inches rotates 6 times a second is:

a. 16π          b.2π                c.12π                        d.none

Solution: c) 12 π. We have: ω= 2π f. The frequency of the tire is 6 revolution per second;

Therefore we can write, ω = 2π × 6 = 12π

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