A particle is moving through what angle does t s angular velocity
Answers
Explanation:
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Centre of mass and Linear momentum
Cross Product of two vectors in Rectangular Coordinate System
Problem on Moment of Inertia
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.
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.