The Big 3 Equations of Motion We have three equations that can be used to solve most Problems, when dealing with translational (tangential) motion. These are the “Big 3 Equations of Motion” Pos f = Pos i + V i t + (1/2)at 2 V f = V i + at V f 2 = V i 2 + 2a(Pos f – Pos i ) Important note: For the tangential motion around the disk the change in tangential position is actually the distance traveled. So if a disk rotates once, the change in tangential position is 1 circumference, even though the change in the Cartesian position is zero.

Rotational motion equations? If there are these equations that can be used to describe the translational motion of a disk, so are there also similar equations to describe the rotational motion of the disk as well? 1 st we know the equations are only mathematical representations of the relationships between position, velocity, acceleration, and time 2 nd we no the relationships between position, velocity, acceleration and time do NOT change when move from a linear system, tangential system, a rotational system, or any other system you can think of. So logically the answer is yes, there should be very similar equations

The logical Angular position equation Pos f = Pos i + V i t + (1/2)at 2 We can start with our linear position equation Which says: End Position = Start position + start velocity * time + half the acceleration * time squared End angular Position = Start angular position + start angular velocity * time + half the angular acceleration * time squared  f =  i +  i t + (1/2)  t 2

The logical Angular velocity V f = V i + at We can start with our linear velocity equation Which says: End velocity = Start velocity + acceleration * time End angular velocity = Start angular velocity + angular acceleration * time  f =  i +  t

We can start with our linear time independent equation V f 2 = V i 2 + 2a(Pos f – Pos i ) Which says: end speed squared = start speed squared + twice the acceleration* the displacement ending angular speed squared = starting angular speed squared + twice the angular acceleration* the angular displacement  f 2 =  i  (  f –  i ) The logical time independent equation

Testing the logic Although logic is a powerful thought process, it does not always work (often due to our limitations, and assumptions) and needs to be validated. To test our logic we can use the our tangential kinematic equations and the equations that relate tangential motion to rotational motion Pos f = Pos i + V i t + (1/2)at 2 V f = V i + at V f 2 = V i 2 + 2a(Pos f – Pos i ) S= Pos =  r v =  r a =  r

Pos f = Pos i + V i t + (1/2)at 2 S =  r v =  r a =  r  f r] =  i r] + [  r]t + (1/2)  r]t 2 S f = S i + V i t + (1/2)at 2  f ] =  i ] + [  ]t + (1/2)  ]t 2

V f = V i + at v =  ra =  r [  f r ]= [  i r ]+ [  r]t V f = V i + at [  f ]= [  i ]+ [  ]t

 f 2 =  i  (  f –  i ) V f 2 = V i 2 + 2a(Pos f – Pos i ) S= Pos =  r v =  r a =  r  f r] 2 = [  i r] 2 + 2[  r]([  f r]– [  i r])  f 2 r 2 =  i 2 r  r 2 (  f –  i ) V f 2 = V i 2 + 2a(Pos f – Pos i )

Validation So in the end the kinematic equations for rotational motion are in essence the same equations that we have always worked with for translational (tangential motion) Translational/ Tangential Motion Rotational Motion Pos f = Pos i + V i t + (1/2)at 2  f ] =  i ] + [  ]t + (1/2)  ]t 2 V f = V i + at [  f ]= [  i ]+ [  ]t V f 2 = V i 2 + 2a(Pos f – Pos i )  f 2 =  i  (  f –  i ) Position equation Velocity equation Time independent equation