Let us look at how we would apply Newton's second law in cylindrical coordinates now cylindrical coordinates.
Is a system that works very handy when you have a particle that moves in three dimensions and where
rotation is involved, so let's say someone sliding down a water slide that spirals down around the
pole, let's say.
So let me explain it this way.
By using this figure, if you have a point and a particle or let's call it a let's call it a body,
because it has mass indicated over here.
Then this particle can move along a curved path in three dimensions.
And we can describe the exact position of this particular making use of three variables, namely Theta,
R and Z.
Now, Theta will give us the angle.
Measured from some fixed line or will give us you can see it as the radius of a cylinder, so the radial
direction, how far out from the center of this particle is situated and then Z gives us the height.
Of this particle, so by specifying a value for Theta, a value for R and a value for Z, we know exactly
where this particle is now.
Theta is given in Radians and then R and Z, usually given in meters.
Now, for each one of these components, we apply Newton's second law.
As we know, Newton's second law states F is equal to M multiplied by eight.
So we are going to have a sum of the forces in this direction.
We're going to have a sum of the forces in the theater direction.
We're going to have a sum of the forces in the radial direction.
So we're going to break down any applied force onto this body.
We're going to break it down in its three components.
And it might be that it only is acting in one of these three or maybe all three at the same time.
And we're going to multiply it by the mass.
And the mass is constant.
We assume the mass to be constant.
So the math doesn't change for the different directions, but the force changes for the different directions.
And therefore, the particle has different accelerations in each one of these three directions.
And it is the formula for the acceleration in each one of these different directions that I'm going
to give you in this lecture.
And if you have that, you can just simply multiply it by the mass to get the force in that specific
direction.
So let's take Newton's second law and cylindrical coordinates, says the sum of the forces in the right
direction, plus the sum of the forces in the theater direction, plus the sum of the forces in this
in the Z direction is equal to the mass multiplied by the acceleration and other acceleration is the
sum of the acceleration in the right direction, plus the acceleration of the theater direction and
the acceleration in the Z direction.
And for each one of these accelerations, we have a different formula.
So four in the radial direction.
We have multiplied by eight where A is given by our double dots.
So the second time derivative of the radius.
Miners are multiplied by Theatergoer Square now DataDot, that is the first time derivative of the theater
angle, so that means it's the angular velocity.
We also call it the angular velocity.
Then in the theater direction we have a multiplied by a C, D as F, so that is multiplied by the acceleration
in the theater direction is given by our theater double dot.
So that is the second derivative of the angle plus two or dot theta dot.
And in the direction we have, the sum of the forces in the in the same direction is mass multiplied
by the acceleration in the same direction that is mass multiplied by Z double dot.
So the second time derivative of the Z coordinate.
So in a question you'll typically be given a function of time for, let's say, the Z coordinates.
You might say that Z increases by the function two T squared.
Right.
And then you can take the second derivative of that and the second derivative of to the squared while
that is four and they have a constant acceleration of four.
So we will do an example where we calculate the different accelerations in the radio, the Theta and
the Z direction, and then multiply it by the mass and then get the forces in each one of these directions.