Question

A particle executes SHM and its position varies with the time as x=Asinwt. Its average speed during its motion from mean position to mid point of mean and extreme position is ?

Answer 1

So you want the average speed. How do you find an average? You need to take all the speeds you want to add up and divide by the total. So what speeds do we need? All of the speeds from the mean position to the mid point of mean and extreme position.

Wow, this problem is so mean it's angry. (arithmetically speaking of course)

So let's break it down into parts, since that's really how you need to approach in. Just do every little bit and identify everything you can and start to piece it together to get what we want. So first off, what's the speed? It's the velocity without a +/- sign attached. So let's get the velocity first, and worry about the sign later. The velocity is just the change in position (x) with respect to time (t) so take the derivative of position!

Now remember, the velocity is constantly changing in SHM since it's going back and forth, so we have to worry about this when we make our average of speeds, since we might be adding up negative speeds and messing up our average. So where are we adding our speeds over? The mean position to the midpoint from the mean position to the extreme position.

The mean position is just the average, which you can calculate easily with this:

\[average =0=\frac{ \int\limits_{t=0}^{t=2 \pi/ \omega} Asin( \omega t)dt }{ \int\limits_{t=0}^{t=2 \pi/ \omega}dt }\]

Since it's adding up every position over one complete period and dividing by the total number of them, which is between 0 and 2pi/w. Remember, integrals are just infinite sums, and since sine waves repeat themselves, we can take the integral over one period to find the average. If none of that makes sense, don't worry, but it's going to come up very soon again. The mean displacement is zero.

So what's the midpoint between the mean displacement and extreme position? Surely you can do this, but remember sine can only ever be +1 or -1 so the extreme is whatever number you multiply that by, which is called the amplitude.

Answer 2

All of the above is great news because that means your velocity isn't going to be going in a different direction over the interval, it has no time to be positive then turn negative, so we can just use our velocity function as is. Now, doing a similar thing as we did above, but with our velocity instead of position, we take the average using integrals.

If you're still having trouble, remember you need the range of speeds that you're averaging.