hey everyone in this video we're going to walk through and solve some example problems using the concepts and equations for rotational motion you can pause the video at the beginning of each problem and try solving it on your own first or you can watch the video and see how we work through these types of problems a wrench is being used to loosen a bolt if the wrench starts out at an angular position of pi over 2 radians and rotates counterclockwise one-third of a revolution what is the final angular position of the wrench in radians so what information are we given we're dealing with rotational motion and the starting position of the wrench the initial angular position is pi over two radians the wrench rotates one third of a revolution which would be a positive angular displacement since it rotates counterclockwise the problem is asking us to find the final angular position specifically in radians here's an example of what this might look like we have a line representing the wrench and it has an initial in your position a final angular position and an angular displacement between them which is pointing counterclockwise since we're working with angular displacement we can use the equation for that Delta theta equals Theta final minus Theta initial we're given Delta Theta and Theta initial so we can find Theta final the problem is asking us to find the final position in radians and the initial position is already in radians so let's convert the angular displacement we write out the value one-third revolutions or one revolution divided by three then we multiply by the unit relationship of two Pi radians per one revolution and we can cross out the Revolutions in the top and bottom multiply our fractions and we get two pi over three radians let's write that in the Box on the left now we can start with our equation remember there are different paths you can take to solve a problem but here we're going to rearrange the equation first to isolate the variable we're looking for Theta final then we can plug in the numbers that we have pi over 2 radians for Theta initial and 2 pi over three radians for the angular displacement when we add those together we find that the final angular position is 7 pi over six radians that seems right we started at a positive angular position then rotated counterclockwise so we added a positive value so we should end up with a positive angle that's greater than pi over two radians if a vinyl record rotates clockwise 1350 degrees in five seconds what is the angular velocity of the record in RPM so first we're given an angular displacement of 1 350 degrees which would be negative because it's clockwise next we're given a period of time five seconds and the problem asks us to find the angular velocity of the record Omega specifically in RPM revolutions per minute here's a rough sketch for this problem we have a record with a line drawn on it we have an angular displacement in the clockwise Direction a period of time and the arrow is labeled with Omega the angular velocity we also labeled the conventional positive and negative directions so we don't forget since we're looking for the angular velocity let's use that equation angular velocity equals the angular displacement over time luckily we're given the displacement and time so it should be easy to find the velocity there's different ways to work through this but we're going to start with this equation and plug in the values that we're given negative 1 350 degrees divided by 5 Seconds when we do the math we find the angular velocity is negative 270 degrees per second however we're not done yet since the problem asked us to find the angular velocity in revolutions per minute so we'll need to convert from degrees to Revolutions and from seconds to minutes starting with negative 270 degrees per second we can multiply by the relationship of one revolution for 360 degrees degrees will cross out and we would be left with revolutions per second so now we need to multiply by the relationship of 60 seconds per one minute the units of seconds cancel out and when we multiply these fractions we get negative 45 revolutions per one minute so the angular velocity of this record is negative 45 RPM does that make sense based on the problem we're rotating some amount in the clockwise Direction which is negative so it makes sense that our angular velocity is negative a lab centrifuge is spinning counterclockwise at 80 radians per second when its speed is turned up if it takes six seconds to reach an angular speed of 450 radians per second what was the angular acceleration of the centrifuge so what can we take away from this problem the object is spinning counterclockwise at 80 radians per second and then later it's spinning at 450 radians per second so we would say that the initial angular velocity is positive 80 radians per second and the final angular velocity is 450 radians per second we're assuming the final velocity is also positive since the problem tells us that the speed increases but it doesn't say anything about changing direction and the period of time it takes to go from one speed to the other is six seconds then the problem asks us to find the angular acceleration there isn't too much we can draw for this one but here we have a round object with the initial and final angular velocities and the counterclockwise Direction and we also noted that there's an angular acceleration in Period of time between the two we're trying to find the angular acceleration so we'll use that equation however like in other problems we're not given Delta Omega the change in angular velocity instead we're given the initial and final values you might already know how to substitute one for the other but we'll write it out in this example the change in angular velocity Delta Omega is equal to the final angular velocity minus the initial angular velocity now we have everything we need to find Alpha the angular acceleration starting with the first equation we can replace Delta Omega with Omega final minus Omega initial and Alpha is already isolated on the left now we can plug in the values we're given for each variable and using a calculator we find that the angular acceleration is equal to 61.67 radians per second squared let's double check that we start with a positive velocity and then increase in Speed without changing direction so we should end up with a positive acceleration like we do here a Carousel Ride is spinning counterclockwise when the operator increases the speed over a period of six seconds the ride accelerates at 2 radians per second squared and rotates through an angular displacement of 40 radians during that time what was the initial angular velocity of the ride so the problem starts by telling us the ride has a counterclockwise so positive velocity but it doesn't tell us what it is for a period of six seconds we have an angular acceleration of two radians per second squared how would we know this is angular acceleration and not tangential first the unit of radians per second squared is the unit for angular acceleration second we're only talking about the Carousel Ride which is rotating not an object on the ride which would be in circular motion next the problem says we have an angular displacement of 40 radians which we assume is positive since the value is positive and we're not told it's in the clockwise Direction the problem is asking us to find the initial angular velocity which we were never given again there isn't too much to draw for this problem but here's something simple so that we can remember we're dealing with rotational Motion in the counterclockwise Direction along with the variables that we're working with we already know we're working with the first constant acceleration equation but if we didn't know that we could look through our list of angular motion equations and see that this equation has everything we need it doesn't have the angular displacement Delta Theta but it does include Theta final and Theta initial and we know that Theta final minus Theta initial equals Delta Theta so we should be able to solve for Omega initial the initial angular velocity let's walk through this problem in two different ways just as a reminder that you can follow the steps in any order first let's rearrange the equations weaving the variables in then plug in numbers at the end we start with our equation for constant angular acceleration if we want to explicitly include Delta Theta in our equation then we first need to subtract Theta initial from both sides moving it to the left then we can replace Theta final minus Theta initial with Delta Theta now we can rearrange this equation to isolate what we're trying to find Omega initial by moving things around and dividing by T now we can plug in the numbers that were given Delta Theta is 40 radians Alpha is 2 radians per second squared and T is 6 seconds using a calculator we find that the initial angular velocity of the carousel was 0.67 radians per second another way we could have done this is to get this equation with Delta Theta and then plug in the numbers there then we can simplify the right side of the equation and then rearrange it to solve for Omega initial plugging this into our calculator is a little easier than before and we still end up with 0.67 radians per second again work through a problem in the order that's easiest for you over time you might prefer one method over the other certain problems might be easier to solve by plugging in numbers early or by leaving everything as variables until the end before we move on let's check our answer the problem says we start with a counterclockwise velocity so our answer should be positive and if we wanted to double check the value remember we could plug in 0.67 radians per second and the given information into the equations that we started with just to make sure the math checks out but we'll move on to the next problem a CD starts from rest and then accelerates reaching an angular velocity of negative 30 radians per second while rotating a total of negative 90 radians what was the angular acceleration of the CD so the CD starts from rest which means it's not moving so the initial angular velocity would be zero radians per second then it accelerates and the final angular velocity would be negative 30 radians per second while it accelerates it rotates through an angular displacement of negative 90 radians and we're asked to find the angular acceleration of the CD here's a quick sketch of the CD showing that it's rotating clockwise because the final angular velocity is negative and we've noted that there's an angular acceleration and an angular displacement while it accelerates we know we're going to use the second equation for angular motion with constant acceleration but like with most problems we could figure that out by looking through the angular motion equations and seeing that this has the variables we're given and the one we're looking for so starting with this equation we could simplify things by Crossing out Omega initial because we're told that the initial angular velocity is zero so we don't have to carry this variable around while we work through the problem at this point we can also replace Theta final minus Theta initial with Delta Theta since we know by now that they're equal to each other now our equation is a bit simpler next we can rearrange the equation to isolate Alpha by swapping the two sides and dividing by two Delta Theta now we can plug in the numbers that we're given calculate Alpha and we get negative five radians per second squared for the angular acceleration of the CD as usual we could plug that back into the equation to check the math but let's just consider the sign the CD starts with zero velocity then ends with a negative velocity so the direction of the acceleration should be negative which is what we got alright thanks for watching and I'll see you in the next video