Why is tangential velocity 2πr/T if the body's total displacement in the end of one revolution is zero?
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5$\begingroup$ Because instantaneous velocity is different than average velocity, and you're thinking about the average velocity over one period instead of the instantaneous velocity at a single time. Even average velocity over a time interval that's not equal to the period is non-zero. This is almost certainly a duplicate here, although so far I've only been able to find a duplicate on Quora. $\endgroup$– marchCommented Feb 27 at 16:26
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$\begingroup$ Okay..but why is the instantaneous velocity using the entire circumference of the circle. Why not just a short displacement with a short time interval ? $\endgroup$– D4NT3 tennysonCommented Feb 27 at 16:53
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$\begingroup$ We're using the entire circle to get the instantaneous speed not the instantaneous velocity. For the speed, you can just compute a total distance traveled divided by a total time, since the speed is constant. The easiest way to do that is to divide the circumference by the period. This yields zero for the average velocity because velocity takes into account the direction of the object and not just how fast it's moving (which is speed). $\endgroup$– marchCommented Feb 27 at 16:58
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$\begingroup$ Do you mean 'Why isn't the speed of Earth around the Sun zero?'? $\endgroup$– my2ctsCommented Feb 28 at 10:15
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4 Answers
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$\frac{2\pi R}T$ is the object's average speed, not velocity. The average velocity is indeed zero, since displacement is zero. If the object is in uniform circular motion, the instantaneous speed is equal to the average speed.
Since distance travelled is not zero, and speed measures distance rather than displacement, it is non-zero.
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$\begingroup$ The argument I saw online was that there are non-zero velocities with no displacement and that's what occurs in circular motion and when you throw an object and it falls down but I still wanted to ask around and get as many perspectives as I can. Thanks $\endgroup$ Commented Feb 27 at 18:38
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$\begingroup$ The instantaneous velocity can still be non-zero, even if the average velocity is zero. Which is what happens in circular motion and throwing an object vertically upwards. $\endgroup$ Commented Feb 28 at 2:32
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Here is an argument based on the definition of vectors (no Calculus).
Velocity is a vector quantity, and hence direction is important. Unlike speed, where direction is not important. When the object is at two different points on the circle, it is going in two different directions, and hence the velocity is nonzero.
Once we have established that the velocity is non-zero, we would like to calculate its magnitude and direction.
Direction The direction of velocity is on the tangent that passes through the circle at the point where the object is.
Magnitude Use the relation between speed and velocity. $$|\vec{v}|=v=Speed$$
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$\begingroup$ Thank you for the reply, but this raises another question. Why do we only consider the magnitude of the velocity and if all we consider is that, why don't we just call it speed and not velocity ? $\endgroup$ Commented Feb 27 at 19:22
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Velocity is directional and in circular motion the velocity is continually changing while the average speed may be constant. Since the velocity is continually changing, we can only get a sensible answer for the velocity at time t, by considering a very small time interval and taking the average of the velocity an infinitesimal moment before instant t and infinitesimal moment after instant t. The magnitude of the velocity at this point is equal to the average speed computed from $2 \pi R /T$ and the magnitude remains constant all the way around but the direction does not. When an object is not travelling in a straight line with constant speed, then Velocity only really makes sense in very small (infinitesimal) time intervals .
Thank you for the reply, but this raises another question. Why do we only consider the magnitude of the velocity and if all we consider is that, why don't we just call it speed and not velocity ?
In the case of circular motion with constant angular velocity, the magnitude of the instantaneous tangential velocity is equal to the tangential speed. This is a special case. If we are considering a particle on the rim of a disk and rate of spin of the disk is increasing, then we can only consider the instantaneous velocity which is continually changing. The same goes for a particle accelerating in a straight line. Physicists generally use the term velocity in preference to speed, as it is more precise and has greater general application than speed.
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$\begingroup$ You are a legend, I wish I could upvote everyone's comment $\endgroup$ Commented Feb 28 at 15:30
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For circular motion, Newton determined that the centripetal acceleration has a magnitude of $\frac {v^2}{r}$ and a direction towards the center. Thus it's velocity not constant.
