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In uniform circular motion, is angular momentum a constant, is that what you're asking? The answer is it depends. Angular momentum, L, is by definition \[ L = r \times p = r \times mv \] This will be constant if you're measuring angular momentum from the center of the circle of motion. But for all other points, it is not.
and angular velocity differ at every point rit
correct, because the vector r changes with the point from which you're measuring it.
i m tired to xplain this thing to arvind he is not listening.. and even accepting !!! well thnx for help he believes u thats y i ask u too ans..
but if r is same ??
now u agree arvind
is angular velocity same?
but do u know its linear velocity??
To be explicit, suppose a body of mass m = 1 kg is moving in uniform circular motion in the xy-plane about the origin at a distance R at an angular velocity \( \omega \). Then the displacement vector r(t) is given by \[ r(t) = R(\cos \omega t, \sin \omega t, 0) \] The velocity is given by differentiating, \[ v(t) = R \omega (- \sin \omega t, \cos \omega t, 0) \] The angular velocity of as measured from the origin is \[ L = r \times mv = r \times v \] \[ = R^2\omega (0, 0, \cos^2 \omega t + \sin^2 \omega t) \] \[ = R^2\omega (0,0,1) \] This is constant.
so angular velocity is constant??
yea by thi it is concluding linear velocity is const. means angular velocity is const. if r is same
Provided the motion is uniform (i.e., constant angular velocity omega) _and_ we are measuring angular velocity from the origin, the center of the motion. As an exercise, calculate L if we measure r from an arbitrary point (a,b). You will see it is not a constant.
wat abu my sentence i wrote i become confuse
Or even an arbitrary point (a,b,c) where this is not = (0,0,0)
by ur exmple
In uniform circular motion, UCM, the vector r is not a constant, nor is the vector v. What is constant is the angular speed \( \omega \).
ok... got it got it thnx i think brent need ur help luk at his
The _magnitude_ of r and v are constant. But the vectors themselves are not constant, as written above.
The magnitude of r, \( |r| = R \), a constant. The magnitude of v, \( |v| = R\omega \), is also a constant.
and w is const for UCM??
Yes, by definition.
i dont have any more medals so that i can give u :(
If \( \omega \) was not constant, then the object would not be moving around at the same rate for all \( t\). But we define UCM to to be the angular velocity \( \omega \) is constant.