Professor Dave here, let's discuss angular motion. We talked about uniform circular motion, but we need to make an important distinction between spin and orbital motion. An object can spin around an internal axis that goes through its center of mass, and it can also orbit around some external axis. This means that the earth will spin on its axis but it orbits around the Sun. We have discussed spinning objects and the way that tangential speed will vary according to distance from the axis of rotation, but this can be a tricky way to view the rotation of something like a ferris wheel, because there is a certain context in which we would like to say that every part of the wheel is spinning at the same speed, since it is one solid object. For this reason when we are looking at a rotating rigid object, rather than looking at translational motion, we will want to discuss angular motion or rotational motion. To do this we can begin to discuss certain angular quantities. Angular displacement represented by theta, will be the angle swept out by any line passing through a rotating body that intersects the axis of rotation. This value will be positive if the motion is counterclockwise and negative if clockwise. If you have a hard time remembering this just remember that in going counterclockwise around the XY plane we will cover quadrants one, two, three, and then four, so positive angular displacement correlates with a positive increase in the number quadrant that is being traversed over time. The SI unit for angular displacement will not be degrees but rather radians. We will discuss the derivation and importance of the radian in the upcoming mathematics course, but for now we simply need to know that one full revolution is equal to 2 pi radians, so if the rotation completed by the object is a counterclockwise quarter turn, the angular displacement is one half pi radians, or pi over two. Angular velocity, as one might expect, is equal to the angular displacement over some time period, just the same way that linear velocity is equal to linear displacement over time. Average angular velocity, represented by the Greek letter omega, is therefore given as delta theta over delta t. The SI unit for this value will be radians per second, although we will frequently encounter other units like revolutions per minute, or rpm. Again, counterclockwise rotation involves positive angular velocity and clockwise will be negative. Lastly, angular acceleration, represented by the Greek letter alpha, is equal to the change in angular velocity over some time period. This will be equal to delta omega over delta t, with units of radians per second squared. Now that we have defined these terms, we can begin to understand that rotational kinematics will utilize equations that are direct analogs of those involved with linear kinematics. We just swap out linear components for angular ones. Looking at these familiar equations, if we just exchange displacement for angular displacement, velocity for angular velocity, and acceleration for angular acceleration, we arrive at a new set of equations that allow us to calculate rotational motion, and they will apply to any system with rotational motion under constant angular acceleration. Depending on the scenario it may or may not be easier to model rotational motion with these equations rather than with the linear ones where we have to use tangential velocities that vary according to the radius. How can we set a system into rotational motion? We can do this by generating something called a torque. Torque is defined as the ability of a force to rotate an object around some axis, whether we are looking at a wheel, a seesaw, or anything else that exhibits this kind of motion, and for now we can just look at a top-down view of a panel attached to a rod allowing for free rotation. Torque, represented by the Greek letter tau, will be equal to the applied force times the distance over which the force is applied times sine theta, where theta is the angle between the force vector and the plane of the rotating object. Torque will therefore have units of Newton meters, and torque will be at a maximum when the applied force is perpendicular to the plane containing the object, because the sine of 90 degrees is one. Any angle less than 90 in either direction will give a sine value that is less than 1, and will therefore diminish the torque, while still producing some motion, until the angle is zero, at which point the whole expression will be equal to zero, and no motion can result. By convention, we will say that torque is positive if it results in counterclockwise rotation and negative if it results in clockwise rotation, as we are by now familiar. This must be the case, because if two different forces act upon this object to produce opposite torques that are equal in magnitude, the object will not move, and the sum of the individual torques must be zero to reflect this fact. We must also note that the magnitude of torque will be equal to the magnitude of the applied force times the length of the lever arm expressed in Newton meters. The longer the lever, the easier it is to produce torque, which is why it is difficult to push a door open if you press near the hinge, but easier and easier as you move outwards, thus extending the lever arm, or the distance from the axis of rotation to the point where the force is applied. This sheds light on the quote by Archimedes, which is generally reported as some variation of the following: give me a lever long enough and a fulcrum to place it on, and I will move the earth. This implies that with a long enough lever arm, one could generate such a massive torque so as to move the world. Wise words from a wise man ,so let's learn about some other things Archimedes said next, but first let's check comprehension. Thanks for watching, guys. Subscribe to my channel for more tutorials, support me on patreon so I can keep making content, and as always feel free to email me:
