00:01
We have some kinematics of motion in a circle.
00:05
So a little bit of a reminder is that we can use rotational kinematics for a particle moving in a circle.
00:14
So there are two kinematic formulas that are similar to linear equations.
00:20
That is, there's a equation that expresses the angular position in time involving both the angular velocity and acceleration.
00:33
And i've left out the angular velocity part in equation one, seeing that my initial angular velocity is zero.
00:43
But i do have an initial position of this automobile starts at an angular position of 180 degrees or pi.
00:54
And this is based off of the directions north, south, east, and west.
01:01
Sort of generalized to be more angular.
01:07
So the car is starting from rest and moving north and curving to the east.
01:17
So i have put it to the left of the origin where the radius of the circle radiates outwards.
01:27
There's a second equation that involves the angular velocity in time and again, i have left out the initial angular velocity since it is zero.
01:40
And finally, there is a centripetal acceleration that links the tangential motion to the centripetal motion.
01:53
So a point of note is the rotational quantities are associated with tangential motion, which means that the particle is moving clockwise or counterclockwise, in this case, clockwise in a circular sense.
02:16
So what we're told is that we have a certain position at t equals 10 seconds.
02:28
That position is an angular position, and we're told a linear velocity, which is related to the angular velocity.
02:39
Through the radius of motion, which we are never given.
02:44
However, both of these pieces of information can be used in order to determine the radius of motion, as well as the constant acceleration.
02:59
So a point of note is that the angular acceleration in this problem is supposed to be a constant, and that means the linear tangential acceleration is also a constant.
03:17
But we're going to set up equation 1.
03:20
So theta final is pi over 2.
03:24
The initial angle position is pi.
03:28
And then we have plus 1⁄2 alpha times 10 squared, which is 100.
03:37
So this is the position and velocity at t equals to 10 seconds.
03:44
We see we can solve that top equation for alpha.
03:48
And it winds up to be minus pi over two, not minus pi over two, minus pi over 100.
04:04
And that should be in units of radians per second squared.
04:10
And we'll get rid of the pie eventually.
04:13
Just when we go to linear quantities, it does not make sense to carry around a pie.
04:19
The omega can then be determined, and that is basically, if we look at its absolute value, because what we want to get out of that is the radius of the motion, and the absolute value of that is the tangential speed divided by the radius, and we do know the tangential speed at 10 seconds.
04:58
It's 30 meters per second.
05:04
And putting all that together, we have pi over 10 is equal to 30 over r.
05:14
And the radius will come out in meters.
05:17
So the radius is 300 over pi meters.
05:25
And now it probably makes sense to drop all the pies.
05:30
We can do a little bit better than that.
05:32
But we have some quantities now that are going to remain constant.
05:38
So we have figured out, and why we need these is we are going to be working with the motion at five seconds.
05:51
So we have figured out so far that the radius is a constant, and it's 300 over pi, which is a little bit less than 100 meters...