hi friends this is not a music class I'm going to use this venile record player to explain the concepts of uniform circular motion in a simple and practical way first a big thanks to the YouTube team for sending us this gift see it has our manocha Academy logo on it now I'm going to place this coin on this record player and let's turn on the record player so when I move the needle here as you can see the dis starts rotating can you see what is the motion of the coin here it is in circular motion and since its speed is constant this is called uniform circular motion so remember a body is said to be in uniform circular motion If It Moves In a circular path at a constant speed what are some other examples of uniform circular motion it can be the the tip of the second hand of a clock the rotating fan blades a Mero round or a satellite orbiting the Earth in all these examples the object is rotating in a circle and it is moving at a constant speed so these are all examples of uniform circular motion to understand and describe uniform circular motion let's discuss some basic concepts like angular displacement angular velocity time period and frequency so let's dive into these Concepts now let's look at our record player as you can see the coin is going around in a circle let's draw a line from the center of the circle to the coin this line is the radius of the circle if we give it a direction we can call this the radius Vector as the body Moves In A Circle the radius Vector will move along with it we can Define the angular displacement of the body as the angle swept by the radius Vector in a certain time interval for example if the body moves from here to here then it has covered an angle of 60° If It Moves a little further ahead the angular displacement is now 90° now if it moves here as you can see the angular displacement is 180° and when it completes the full circle what will be the angular displacement of this coin that's right 360° you can measure angular displacement in degrees but the SI unit is radians and you know the relation between degrees and radians that's right 360° is 2 pi radians so for a 90° angular displacement the displacement in radians is going to be Pi by 2 radians for a 180° displacement it's going to be Pi radians and for 360° the full Circle the angular displacement will be 2 pi radians angular displacement is a vector quantity but what is its direction note that it is not along the angle angular displacement is an actual Vector its direction is along the axis passing through the center of the circle but is the direction outwards or inwards you can easily find this by using the right hand thumb rule just curl the fingers of your right hand in the direction of the motion and the thumb will give you the direction of the angular displacement and that reminds me don't forget to hit the like button for this video and share it with your friends all right let's continue on the right hand thumb Rule and use it to find the direction of the angular displacement of this coin so as you can see the coin is moving around like this and if I use my right hand I need to curve curl my fingers in this direction of the right hand right and where is my thumb pointing as you can see the thumb is pointing downwards so basically the direction of angular displacement is going to be inwards over here into the circle this way how is angular displacement connected to linear displacement linear displacement means the displacement along a line so here as the coin moves it is the displacement along the tangent to the circle now from mathematics we know that angle equals Arc / radius if the angle is denoted as Theta Arc as s and radius as R we get the formula Theta equal s / R so angular displacement is linear displacement divided by the radius and if you rearrange this formula you will get s = r Theta this expression tells us that if the angle is the same as the radius increases the displacement will increase you can practically see this if I place another coin at a different distance from the center of the circle as the record player rotates what is the angular displacement of both these coins that is the red and yellow coin that's right it's exactly the same because both coins are covering the same angle in the same time so they're having the same angular displacement but what about the displacement or the distance covered along the circle clearly you can see that the yellow coin is covering a smaller distance since it is closer to the center and the red coin which is further away from the center is covering a much larger distance because the radius is more for the yellow coin the radius is smaller for the red coin the radius is larger so therefore the red coin is having a larger linear displacement and this is exactly what we expected because the formula is s = r Theta so greater the radius more will be the displacement so you can see the red coin is covering a much larger distance as compared to the yellow coin the yellow coin rotates in a small circle but the red coin is rotating in a much larger Circle even though both are having the same angular displacement so remember the relation s equal to R Theta here Theta is the angular displacement its unit is in radians and it is an axial Vector passing through the axis which is the center of the circle here and S is the linear displacement or the tangential displacement along the tangent to the circle and they are related with s = r Theta where R is the radius of the circle so remember this important formula s = r Theta it is connecting the angular displacement with the linear displacement we know that a body in uniform circular motion has an angular displacement and since the body is in motion it will have some velocity that is angular velocity now how do we Define angular velocity remember velocity is the rate of change of displacement so similarly angular velocity becomes the rate of change of angular displacement velocity the formula is displacement divided by time so similarly angular velocity the formula is going to be angular displacement divided by time and we denote that as Delta Theta by delta T Delta Theta is the angular displacement and delta T is the time and angular velocity is denoted by the symbol Omega we use this Greek symbol and the lowercase symbol Omega here so Omega equal to Delta Theta by delta T that's your angular velocity formula now can you tell me what is the unit of angular velocity angular velocity is angular displacement divided by time so the unit is going to be radians divided by seconds that is radian per second now let's say our coin here covers 360° that is 2 pi radians in 2 seconds so what will be its angular velocity you can use the formula Omega = Delta Theta by delta T now substitute the Delta Theta which is the angular displacement as 2 pi radians and the time to complete this motion to complete the circle is 2 seconds so it's going to be 2 piun / 2 which is piun radian per second that is the angular velocity of this coin or if you want the numerical value that's going to be 3.14 radians per second now note that this is the average angular velocity Omega equal to Delta Theta / delta T but if the time interval delta T is very very small that is in the limit delta T tending to zero our formula becomes Omega equal to D Theta by DT using calculus and this is the instantaneous angular velocity that means the angular velocity at any particular instant so that is Omega equal to D Theta by DT we have to do the differentiation with time here so remember the average angular velocity formula is Delta Theta by delta T and the instantaneous one is D Theta by DT time period is defined as the time taken by the body to complete one revolution along the circular path this coin here is taking 2 seconds to complete one full Revolution so what is the time period the time period is going to be 2 seconds time period is denoted by capital T and its unit is seconds now what is frequency frequency is the number of revolutions completed by the body in one second or we say in unit time frequency is denoted by F or you can use the Greek letter new I like to use f for frequency it's simpler now what do you think is going to be the frequency of this coin since it takes two seconds to complete one revolution so in 1 second it's going to complete half a revolution so the frequency is half the SI unit of frequency is 1 by second or denoted as Herz so frequency of this coin is half Hertz or 0.5 Hertz so remember time period is the time taken to complete one revolution and frequency is the number of Revolutions in one second so time period and frequency are the reverse of each other and the relation between time period and frequency is T = to 1x F or F = 1X T So as you can see they are reciprocals of each other how are angular velocity time period and frequency related it's very simple to derive the relation start with the formula of angular velocity we know that angular velocity is angular displacement divided by time now if you take the time as the time to complete one full Revolution then the time will be the time period capital T because it is the time taken to complete one full Revolution and what will be the angular displacement in one full Revolution that's right it's going to be 2 pi radians because you're completing the entire circle so what is the angular velocity the angular velocity is going to be 2 Pi / T so that's our important relation Omega = 2 pi by T and remember 1 BYT is the frequency so we can also write angular velocity Omega is 2 pi f 2 pi * the frequency so see it's so simple to derive the relation when a body is in uniform circular motion we talked about two displacements linear displacement or tangential displacement that is measured along the tangent to the circle and angular displacement that is the angle covered similarly we can have two types of velocities we can have linear or tangential velocity and angular velocity let's look at what is the relation between linear velocity and angular velocity linear velocity or simply velocity is the rate of change of displacement linear velocity the formula is V = to DS by DT remember linear velocity is along the tangent to The Circle at a point so we can also call it the tangential velocity and angular velocity our formula was Omega = to D Theta by DT now how are these two velocities related to derive the relation let's start with the formula angle equal to AR by radius so remember we talked about it that Theta is s / r or Delta Theta is Delta s by R now if we divide both the sides of these equation by the time delta T we are going to get Delta Theta by delta T = to 1x R * delt s by delta T now now in the limit delt T tending to zero we get D Theta by DT = 1X R * DS by DT so angular velocity Omega is going to be 1X R * the linear velocity V or you can write it as v = r Omega so linear velocity equals radius time angular velocity so larger the circle that is for a larger radius linear velocity will be larger as you can see with these two bodies placed at different distances from the center of the circle both the bodies are covering the same angle in the same time so their angular velocity is the same now is the linear velocity of the two bodies same no as you can see the yellow coin appears to be moving much slower compared to the red coin the red coin is moving much much faster with a greater linear velocity why because it is covering a larger Circle so larger the radius obviously the linear velocity is going to be larger since angular velocity linear velocity and radius are all vectors we can use the vector notation in the formula and write it like this we had similar relations for displacement and velocity remember we saw s = r Theta and here we have V equal to R Omega so in both these equations we are multiplying with the radius of the circle R so it's easy to remember s = r Theta and V the linear velocity equals R * Omega the angular velocity is uniform circular motion an accelerated motion what do you think we are tempted to say no since the body is moving at a constant speed so acceleration should be zero right remember acceleration is defined as the rate of change of velocity velocity is a vector quantity it has both magnitude and direction in uniform circular motion even though the speed of the body remains constant the direction of the body is constantly changing at a particular instant the direction of the Velocity is a tangent to the circle and as the body moves in C circular motion the magnitude of the Velocity remains constant but the direction of the Velocity keeps on changing so the velocity is constantly changing now since the velocity is changing the body must be in acceleration so uniform circular motion is an example of accelerated motion now what is the value and direction of the acceleration here since the tangential velocity is constant in uniform circular motion the the acceleration cannot be in that direction that is it can't be in the tangential Direction the acceleration has to be perpendicular to the tangent that means perpendicular to the tangent to The Circle at any point in the body's motion if the acceleration is perpendicular to the tangent that means it is along the radius of the circle acceleration is always along the radius towards the center of the circle since the acceleration is always Center seeking it is called centripedal acceleration because the word centripedal means Center seeking Now by using some vector addition and similar triangles in Geometry you can work out the formula for the centripedal acceleration here you can refer to your textbooks for the detailed derivation and finally you will arrive at this formula AC equal to v² / R I'm using AC over here to denote that it is centripetal acceleration now if you substitute v = r Omega in the equation AC equal to v² by R then you'll get another formula AC = Omega s r so both formulas are correct in one we are expressing centripedal acceleration in terms of the tangential velocity V and in the other one we are expressing it in terms of the angular velocity Omega so remember these two formulas centripetal a acceleration is denoted by AC in some books you'll find it denoted by a r here R stands for the radius because it is the radial component of the acceleration it's along the radius and Center seeking so remember to write down and remember these formulas of centripedal acceleration so uniform circular motion is definitely an accelerated motion till now we have discussed uniform circular motion where the body is moving in a circle at a at a constant speed but what if the body is moving in a circle at a variable speed take the example of a rotating fan if you consider the blade of the fan it is initially at rest and when you switch on the fan it starts rotating and slowly gains speed right so it is definitely accelerating is the angle of velocity of the fan increasing in this case the answer is yes because when you switch it on it goes faster and faster so the angular velocity is definitely increasing this means that there must be some angular acceleration now if you consider the tip of the fan that has just been switched on the tip of the fan is also moving faster and faster so there is linear or tangential acceleration as well so let's look at angular and linear acceleration for circular motion and what is the relation between the two we we know that acceleration is the rate of change of velocity similarly angular acceleration is the rate of change of angular velocity angular acceleration is represented by the symbol Alpha so we can write the formula angular acceleration Alpha is Delta Omega by delta T where Delta Omega is the change in angular velocity and delta T is the time so Alpha is Delta Omega by delta T this is the formula for average angular acceleration now as the time interval delta T becomes really really small that is the limit delta T tending to zero we get instantaneous acceleration that is Alpha equal to D Omega by DT now what is the unit of angular acceleration it's going to be radians per second squar now let's take a look how angular acceleration is related to the linear acceleration or you can call tangential acceleration for that we will start with the equation linear velocity equals radius time angular velocity or V equal to R Omega now differentiating both the sides of the equation with respect to time we get DV by DT equal to DDT of R Omega since the motion is in a circle the radius R is constant so we can bring it out of the differentiation here so we get DV by DT equal to R * D Omega by DT now DV by DT is the rate of change of linear velocity so that is linear acceleration and D Omega by DT is the rate of change of angular velocity so that's going to be angular acceleration so as a result we get a equal to R * Alpha so linear acceleration is R * the angular acceleration and in Vector notation you can write it as a = to R * Alpha with the arrows on top of the quantities since they are representing vectors so remember the important relation that we saw linear acceleration is radius times the angular acceleration so in a circular motion there can be three types of acceleration angular acceleration centripedal acceleration and linear or t T ential acceleration angular acceleration is the rate of change of angular velocity and remember it's an axial Vector it is along the axis of the circle and it is in the same direction as the angular velocity the formula for angular acceleration is Alpha = to D Omega by DT its unit is radian per second squar centripedal acceleration on the other hand is directed towards the center of the circle C and its formula is AC = v² by r or AC = Omega s r linear or tangential acceleration is the acceleration along the tangent to the circle to make it clear we can denote it as a so that you don't confuse it with the other accelerations and the formula of at was R * Alpha the angular acceleration so = R Alpha when the body is moving in a circle and accelerating it will have two accelerations in the plane of the circle one is towards the center of the circle which is centripedal acceleration and the other is tangential to the circle that is tangential or linear acceleration now these two accelerations are perpendicular to each other both these accelerations centripedal acceleration and tangential acceleration have the units of m/s squ so we can find the net acceleration by adding these two vectors and the magnitude of this net acceleration will be a net equal to square < TK of a² plus AC s we are basically using the vector addition and Pythagoras Theorem here because both the accelerations are perpendicular to each other so this was for circular motion and and note that when a body is in uniform circular motion that is when the body is moving at a constant speed angular acceleration and tangential acceleration are both zero in uniform circular motion the only acceleration is centripetal acceleration the acceleration towards the center of the circle since the direction of the Velocity is constantly changing but the magnitude of the Velocity is constant so you only have centripedal acceleration in uniform circular motion I hope the concepts of uniform circular motion are crystal clear to you now now let me ask you a question let's say we have a body of mass 5 kg which is revolving in a circle of diameter 14 cm at 300 RPM that is 300 revolutions per minute now I want you to calculate the linear velocity and centripedal acceleration experienced by this body do let me know your answers by putting it in the comments below I promise to reply to your answers so do try this question and put in the answers in the comments below so the next time you're sitting on a merry round or you're watching the second hand of the clock ticking remember the concept of uniform circular motion and if you like this video hit the like 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