find the vector relation between linear and angular velocity
Answers
What is relation between linear velocity and angular velocity?
Relation between linear velocity and angular velocity
Let us consider a body P moving along the circumference of a circle of radius r with linear velocity v and angular velocity ω as shown in Fig.. Let it move from P to Q in time dt and dθ be the angle swept by the radius vector.
Let PQ = ds, be the arc length covered by the particle moving along the circle, then the angular displacement d θ is expressed as dθ = ds/r. But ds=vdt.
d θ/dt=v/r
(i.e) Angular velocity ω = v/r or v =ω r
In vector notation,
Vector v = Vector ω x Vector r
Thus, for a given angular velocity ω, the linear velocity v of the particle is directly proportional to the distance of the particle from the centre of the circular path (i.e) for a body in a uniform circular motion, the angular velocity is the same for all points in the body but linear velocity is different for different points of the body.
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Originally Answered: What is the difference between angular velocity and linear velocity?
Linear velocity is simply how fast, and in what direction an object is moving (in SI units this is expressed in metres per second) relative to some frame of reference. Something which is moving in a straight line at constant speed is said to be in an inertial frame of reference (that is there is no acceleration).
In contrast, angular velocity is the rate at which an object is rotating around an axis (angular displacement over time). It is often expressed in radians per second although it could equally be expressed in degrees or total revolutions in a time period. Indeed the units can be reversed - we might say that the Earth orbit the Sun once every three hundred and sixty-five and a quarter days. (nb. a pedant will point out that the Sun and Earth actually rotate around a common point, but as that common point is well within the bounds of the Sun, it is usual to describe the Earth as orbiting, or going around the Sun).
What is also inherent in any object rotating around an axis is that it is undergoing a continual acceleration towards that axis. If (like the moon) that acceleration towards the centre is such that (over time) it will maintain much the same distance from the point it it rotating around, it can be said to be in orbit. If the acceleration is insufficient it will fly off in some other direction. Something which has angular momentum is inherently in a non-inertial frame of reference (but see later).
A real-world object with angular velocity will also have linear velocity. For example, an old fashioned vinyl LP will rotate at thirty-three and a third revolutions every minute, but the edge of that record will have an instantaneous tangential linear velocity of about 0.5 m/s.
In purely mathematical terms it is possible to have a rotating zero-dimensional point which doesn’t have a circumference speed, but that’s not really something that impacts on everyday physical objects, but it has a deeper meaning in physics and can be applied to things like the changing electromagnetic fields in light. It also pervades quantum mechanics and much else.
One curious feature about angular velocity is that it can be considered to be absolute. That is an object can be said to be rotating around an axis without reference to any other object. A person placed in an enclosed box could instantly tell if it was being rotated around a point due to that acceleration towards the axis. However, that person would have no idea if they are travelling linearly (in an inertial frame). So an inherent difference between linear movement (an inertial frame) and an rotating frame (a special case of a non-inertial frame) is that the former is relative whilst the other is absolute, at least using our normal understanding of space and time and even in Einstein’s Special Theory of Relativity.
Intriguingly, with Einstein’s General Theory of Relativity (GToR) he managed to effectively eliminate the difference between inertial and non-inertial frames of reference, at least in terms of gravitational effects. So, in some sense, that difference between linear and rotational velocities becomes somewhat slippery once you start getting into the esoteric world of the GToR.
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Linear velocity is the tangential component of velocity of the rotating mass while the angular velocity is the rate of change of angular displacement. Angular velocity is a vector that is perpendicular to the plane of rotation of the rigid body.
Get a better sense of this in this video
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Originally Answered: How does linear velocity relate to angular velocity?
Let θ be the angular displacement of the body.
Let "t" be the time taken.
Then,
ω=θ/t
For a very small value of "t"
ω=Δθ/Δt
Now let,
S=r.θ(where S is distance covered/Arc length and r is radius of the circle/circular path)
→ΔS=r.Δθ
→Δθ=ΔS/r
Then ω=Δθ/Δtcan be written as,
→ω.r=ΔS/Δt
→V=ΔS/Δt
→V=ω.r
V=ωr
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Explanation:
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