in this problem i'll talk about spinning bicycle will so we have a cyclist here and she is the cycling and then the wheel when she starts doing it the cycling it the wheel starts from rest and it comes to a final angular speed of 200 rpm the rpm stands for the revolution per minute and the time it takes is four seconds and now we need to calculate what is the average angular acceleration that's the first part and in the second part what you have to calculate is now see in the second part she applies the brake after she gets tired she applies the brake and the wheel comes to stop in 20 seconds now you need to find out again the average angular acceleration and what is the angular displacement and also you have to find out the number of revolutions or how many turns the the bicycle will max it before coming to rest so let's do it so first part i'm just talking here about the first part here as the wheel is starting from the rest its initial angular speed is zero omega is the angular speed and i stands for the initial which is zero radian per second and it comes it does have an angular speed of 200 revolution per minute in four second so the final angular speed again f stands for final is 200 evolution per minute you need to change revolution per minute in radian per second always remember to change the angular speed to radian per second how do you change it so it's given here rpm revolution per minute to radian the one revolution that's the first revolution per minute a one revolution what is the one revolution means this means one complete cycle which is which has an angle of 2 pi or 360 so 2 pi radian one revolution again means 2 pi radian and the 1 minute is 60 second so 260 you'll get 30 here so one revolution per minute is pi over 30 radian per second this is three zero okay so now it's easy to change it so simply you have to multiply by pi over 30 so 200 now pi over 30 that give takes you to radian per second so after calculating what i got is my final angular speed is 20.9 radian per second and the time is also given is four seconds that's the final time and the yo you are starting your starting time is already zero or if it is given at the time that you can choose that time so this is four second or the change in time basically is four second so we'll calculate the average angular acceleration as it says the angular acceleration or average angular acceleration what does that mean is what is the change in a speed divided by the change in the time okay so you need to find out the change in the angular speed so this is the change in the angular speed and this is the time or the change in time or the time interval so the change means the final angular speed minus initial angular speed and the final time minus the initial time and the final angular speed is 20.9 and as the wheel is starting from rest its initial angular speed is zero and again the final time is four second and you are starting your time from zero so this is your what you get is five point two two radian per second and this is a positive value what is the positive value of angular acceleration means the positive value of angular acceleration means that the speed increases the wheel is turning faster and faster that's what the positive value of angular acceleration means so if you need to understand a little more about it the average angular acceleration you need to you already have studied about the average linear acceleration and it's exactly the same thing this is in linear motion so average angular acceleration is the change in velocity divided by the change in time that's exactly what it is here so you just need to translate the linear motion to rotational motion that's it and one thing i'll just make a slightly change as it is saying the average i'll just put one bar here that means this is the average so this line this lines always means the average okay and the second thing we need to calculate is now okay so now c applies the break and then we need to again calculate the average angular acceleration so before applying the brake the the wheel was turning the wheel was turning and the angular speed at the time was 20.9 that's for the part one now for now for this part once she applies the break this becomes the initial angular speed 20.9 and she has applied a break and after certain time it comes to rest or it comes to stop so the final angular speed will be zero and this will be initial angular speed okay for the second part when she applies the prey so you just see the notation now for the when she was speeding up when she started cycling so the her initial angular speed was zero now she applied the brake her final angular speed is zero but now her initial angular speed is the the final angular speed of the uh when she was cycling or she started cycling or she started speeding up so the final time is 22nd because it says it takes to 20 seconds and now the change again the same formula here the average angular acceleration the average angular acceleration is now a change in angular speed divided by the change in time which is the final angular speed minus initial angular speed what is the final angular speed which is zero radian per second and the initial angular speed or omega naught sometimes we also denoted by omega i is exactly the same thing but just to make consistency or i'm just calling this an omega i it's exactly the same thing so some books just prefers to write omega 0 sometimes even i write omega 0 omega i so it's exactly the same thing omega f which is the final angular speed is 0 because it finally came to stop and this is your initial 20.9 and the change in time is is 20 seconds now if you solve it what you get is negative 1.05 what is the negative sign means the negative sign means the angular speed is decreasing that means the wheel or the bicycle wheel is coming to a stop that's what the negative sign means if it is positive what does that mean that means it is speeding up the negative sign means it is slowing down and now we need to find out how many what is the angular displacement or how many turns the wheel has turned before coming to a stop so let's calculate it so again all the formula here if you know the all the formula in linear dynamics and in linear motion you can simply translate into the rotational motion so again the angular displacement you you need to familiarize familiarize yourself with all the notations so the angular displacement here is simply equal to theta okay we know the this equation v squared is equal to u squared plus two airs and this is in linear motion and i'm going to translate this equation into rotational motion so v is the angular velocity so in terms of the rotational motion it will be angular speed omega f and u is the initial velocity so it will be initial angular speed plus 2 a is the angular is the linear acceleration and in linear motion it will be angular acceleration and s is the the distance the linear distance and in rotational motion it will be the angular distance or what you call the angular displacement that's as simple as it is so the final angular speed is zero because the wheel has finally come to a stop and the initial angular speed is 20.9 that's already we have calculated this one 20.9 plus 2 alpha which is the angular acceleration we just calculated is 1.05 and theta we need to calculate if i solve for theta theta is 208 radian so sometimes we are not very used to with the regions so let's calculate the number of revolutions so one revolution is equal to 2 pi radian that means if it makes one complete rotation the wheel when one complete revolution the total angle is 2 pi to the 2 pi radian so 1 radian is 1 over 2 pi revolution as this is in the given in radian we need to change the radian into revolution that means we simply need to divide it by 2 pi so 200 hpn and dividing by 2 pi that gives me a revolution so the wheel has come made a complete turn of 33 complete turn before coming to a stop so this is how you calculated the average angular acceleration and the angular displacement the number of revolutions made for a spinning bicycle wheel again if you have any questions write down your questions in the comment section below and do not forget to like share and subscribe the channel thank you very