in this video we're going to talk about rotational motion so what exactly is rotational motion from the word you can describe that it's any object that can rotate or spin so that's rotational motion it's different from linear motion where an object simply moves forward linear motion and translational motion basically means the same thing now so what terms do you need to be familiar with when dealing with rotational motion well in linear motion you have things like position and displacement for instance let's say if this is a number line and an object is currently at position a so position is basically a point in space displacement is the change in position let's say if the object moves from x8 to xb then the object's displacement is the difference between xb and x8 it's the final position minus the initial position that's displacement well for rotational motion we have the same terms there's angular position and angular displacement so angular position just like regular position is a point on the circle let's call this point a and let's say as the wheel spins point the object moves from point a to let's say point b so this angle represents the angular position and this is also the angular position but the difference between those two points i guess you can call it delta theta this is the angular displacement is the difference between those two points in a circle it's the final angular position minus the initial angular position so that's angular displacement now the standard unit for angular displacement is the unit radians now you can use degrees or any type of angle measure but the most common unit that you're going to see is radiance now the next term that you need to be familiar with is angular velocity angular velocity tells you how fast an object is spinning on a circle whereas linear velocity tells you how fast it's moving forward so the average linear velocity or let's say average linear speed is distance divided by time velocity is really displacement over time but the units will still be meters per second now angular velocity which is represented by the symbol omega this is going to be average angular velocity it's equal to the angular displacement divided by the time and as you can see the equations are very similar so for average velocity we have units such as meters per second and for average angular velocity it's radians per second so here this is going to be displacement over time to get velocity and to get angle velocity its angular displacement over time now the equation that connects linear velocity to angular velocity is this equation linear velocity is equal to omega times r so whenever you have an object spinning the angular speed is the same everywhere on a circle however the linear speed is not the same so let me give you an example that will illustrate this so let's say this is the center of the circle and this circle is spinning at 5 radians per second so let's say this is point a and this is point b let's move point b to this position so which one has the greater angular velocity point a or point b now because the wheel is spinning at one angle speed every point on this will will travel at that same speed it's going to rotate at that same rate so the angular velocity of a is equal to the angular velocity of point b they're going to rotate at the same rate now what about the linear velocity is it the same or is it different now they all spin at the same rate but point a has a shorter distance to travel point b has to travel a longer distance and in order to spin at the same rate b has to travel faster than a so therefore the linear speed of point b is greater than the linear speed of point a and there's an equation that we can use it's this one v is equal to omega times r so as r increases the linear velocity will increase so notice that our a is less than rb so that's one reason why the linear velocity of b is greater than the linear velocity of point a now there's some other terms that you need to know and that is the period and the frequency the period is the time it takes to complete one cycle so if you have the total time and the number of cycles that will make or rotations if you divide these two you're going to get the time it takes to make one complete cycle or one full rotation around the circle frequency is the reciprocal of period it's the number of cycles that occurs per second so if you have the number of cycles and you divide it by the time this will give you the frequency so the unit for period is the seconds but it's really seconds per cycle or seconds per revolution the unit frequency is hertz or one over seconds now period is one divided by the frequency can also calculate the angular velocity if you know the frequency the angular velocity is equal to 2 pi f so this is one equation that you want to commit to memory and you also want to know this one as well now if you have the period and you want to calculate the angular speed you can use this equation omega is equal to two pi divided by t so make sure you know those three equations that can relate angular velocity with period and frequency now the next thing that we need to talk about is angular acceleration and linear acceleration so linear acceleration is the change in velocity divided by the change in time so this is average linear acceleration the average angular acceleration is equal to a very similar equation it's the change in angular velocity divided by the change in time the units for linear acceleration is meters per second squared for angle acceleration it's radians per second squared at least that's the standard unit it's rare that you'll see degrees per second squared but that would still be a form of angular acceleration but this is the most common unit now whenever you have an object moving in circular motion there's different types of accelerations that you need to be familiar with so if the object has any kind of speed if it's moving around a circle it's going to have a centripetal acceleration and that centripetal acceleration points towards the center of the circle the centripetal acceleration is equal to the linear speed squared divided by the radius of the circle and we know that v is equal to omega r so the centripetal acceleration is going to be omega r squared divided by r so you can cancel one of the r's so just to take it one step at a time once you square omega r you're going to have omega squared times r squared which is r times r and then you can cancel one of these so the centripetal acceleration is omega squared times r so that's the acceleration that points towards the center of the circle so if an object is moving with constant speed around a circle the only acceleration it has is the centripetal acceleration also known as the radial acceleration so the net acceleration of the object will equal ac if it's moving with constant speed now what if it's not moving with constant speed what if it's accelerating around a circle so then it's going to have a tangential acceleration and the tangential acceleration is equal to the angular acceleration times r and we also found the tangential acceleration using this equation it's the change in velocity divided by the change in time now notice that these two vectors they're perpendicular to each other so if we have the centripetal acceleration going this way and this is the tangential acceleration therefore the net acceleration of the object at this instant is going in the northwest direction so when the object is not moving with constant speed around a circle the net acceleration is the vector sum of the centripetal acceleration and the tangential acceleration so basically it's the hypotenuse of the right triangle you