in this video i'll talk about the rotational dynamics of a centrifuge a centrifuge is simply a rotating device or is a spinning disc the information that is given is the the radius of the spinning disc or the centrifuge which is 25 centimeter and it rotates clockwise so that means this direction this machine is turned off it comes to a stop and from its maximum rotation or 4000 rpm that means 4 000 revolution per minute and it takes 30 seconds to come to stop now you need to find out the average angular acceleration the tangential acceleration the net acceleration the total acceleration okay so here we might be using a lot of terminology the same kind for example angular acceleration the tangential acceleration and what are the speed the linear speed the the centripetal acceleration so do not get confused okay and i already have made another videos about all this terminology so please watch those videos as well so let's start now the radius is 25 centimeter the initial angular speed omega i stands for the angular speed which is 4000 rpm revolution per minute what does this mean revolution per minute the 4000 revolution per minute means in one minute or in 60 seconds this centrifuge machine turns 4 000 times or it makes 4 000 complete turn in one minute and we need to change whenever you're going to use the dynamics equations then you make sure that your angular speed is in the unit of radian per second so this is the unit you need to put in so we have to change revolution per minute to radian per second how do we do that so here is the formula rpm to radian one rpm is the revolution per minute and one revolution is 2 pi radian that means if it makes one complete rotation or one complete evolution then the total angle is 2 pi radian and 1 minute is 60 seconds so 2 over 60 is 30 so in order to now change from rpm to radian you have to simply multiply over pi by 30. so here now i'm multiplying pi over 30 and it will take me to radian per second which is 419 radian per second and the final angular speed is 0 which makes sense because it has finally come to a stop and the time to come to stop is 30 seconds so we're going to calculate the angular acceleration and this is average angular acceleration so how do you calculate the average angular acceleration the way you calculate its change in the angular speed divided by the time that's what it means the change in the delta the this triangle is the delta omega divided by the delta time which is the change in the time the final angular speed is zero and the initial angular speed is 419 and the total time is 30 second if you do the math what you get is negative 13.97 what is the negative sign means i would like you to pause the video and think about what does the negative sign mean the negative sign simply means this a spinning disc is slowing down or it is coming to a stop that's what what the negative sign means so this is the average angular acceleration now let's calculate the tangential acceleration in order to calculate the tangential expression the tangential acceleration is simply r times alpha and r is the radius so if you simply multiply this angular acceleration by the radius you'll get the tangential acceleration it is asking you to calculate at 20 centimeters i think this has to be 20 25 25 centimeter so at 25 centimeter away from it so this is what you get 3.49 the one thing now you need to notice is the tangential acceleration is the distance dependent it depends upon the position where you are at this point or this point or this point the further you are the more the tangential acceleration you have but the angular acceleration it doesn't matter where this point this point this point or this point it is exactly the same but the tangential acceleration is dependent on the position the further away you are the more the tangential acceleration you will have okay so if you need to think about the let uh differences between the tangential and angular acceleration so the tangential acceleration is always at the tangent okay and the angular acceleration the direction depends upon whether it is speeding up or it is slowing down now let's say there's a spinning disc here this is spitting disc is turning clockwise so you have to use your right hand curl rule and the right hand curl rule tells you let me just show you one second let me bring out one thing here so what is the right hand rule tells you is if you curl your finger just like this if you curl your finger in the direction of the rotation then your thumb points out the direction of the angular speed okay so that's the thing you need to keep in mind so in this case if i turn my fingers in the direction of the rotation my thumb points downward or in other words it points downward this way that's the direction of the angular speed but my angular acceleration will be upward why is that why how come the my angular acceleration will be upward and the reason is because the spinning disc is slowing down if a things is slowing down or if a spinning disc or a fan or a turbine or a generator is slowing down the angular speed and the angular acceleration will be in opposite direction okay so the angular acceleration remember the angular acceleration can be either upward or downward and the tangential expression is a tangent at the different points okay so that's a short explanation about the different things and watch my other videos to get a clear explanation about all this terminology okay so now let's calculate the angular speed the angular speed is omega time is equal to omega i plus alpha t and this is the final angular speed this is the initial angular speed alpha is the angular acceleration and t is the time and we are asked to calculate angular speed at 29 second after the machine is turned off so the time is 29 seconds and we know that the initial angular speed is 419. the alpha we just have calculated age here 13.97 then the time is 29 so if you plug in what you get is 13.87 that's the angular speed now let's calculate the tangential speed the tangential speed would be again r times omega that's the formula for calculating the tangential speed or if you simply multiply this by r that gives you the tangential speed and the tangential r the distance is also given here it says the 25 centimeter so the 25 times 13.87 if you multiply you'll get 3.47 meter per second just take a note here so the angular speed has a unit of radian per second and here it has a unit of meter per second remember the angular speed the direction again can be either upward or it can be downward but the linear speed has either this at the tangent or the tangent okay so we now calculated the tangential speed but our goal is to calculate the another acceleration that is the centripetal acceleration so when when an object is moving in a circle then it does have many acceleration the one acceleration is the tangential acceleration and this tangential acceleration comes only when it is not moving with a uniform speed if it is slowing down then it will have opposite to the velocity the part the this disc is spinning this way but but as it is slowing down the tangential acceleration is opposite but the centripetal acceleration is always towards the center okay so the net acceleration will be at some other angle all right okay so the the formula for calculating this centripetal acceleration is vt square over r and this is the tangential velocity or you might have studied about the centripetal force which is m v square over r so this is the acceleration v square over r that's exactly what it is here and the tangential acceleration we just calculated 3.47 and the radius is 0.25 that gives you 48.1 that's the centripetal acceleration so the direction is appropriately shown here this is the centripetal acceleration and this is the tangential acceleration so net acceleration is at some other angle so how do we find out the magnitude of the net acceleration it can be given by the square root of ac squared plus 80 squared you can clearly see this is a right angle triangle if i just look at this way so this is the net acceleration and it will be a square of this term the square root of this which is the tangential acceleration and then take the square root that's how you find the hypotenuse so ac squared plus 80 square ac is 48.1 and a t is negative 3.49 and what you get is close to 48.2 so the net acceleration is still close to this centripetal acceleration and the direction the direction this sin this the total acceleration is not radially inward because it has a small component the tangential component the way we can find it out is the tangent theta is ac over a t so if you have something so it's like theta here then rise over run and gives you the the angle here so in this case in this case if i look into this triangle here the rise will be ac and the run will be a t because i'm taking this angle a ac over a t the tangent theta and if we calculate and you'll get 85.9 degree again the negative sign simply means it is opposite to the the motion here okay so this gives you the the direction and that's it for the centrifuge net acceleration so again if you want to understand all the terminal logic if you are not familiar watch my other videos and if you have any questions any sort of questions write down your questions in the comment section below and do not forget to like share and subscribe the channel thank you