So let us look at an example where we apply Newton's second law by using cylindrical coordinates.
Now, what I've illustrated in a figure here is a tennis ball with a weight of zero point zero five
six kg, which is swung around a ball.
The string to which the tennis ball is attached slides down the pole.
Such that its position in the Z direction is given by miners to T meters, so you can see it's negative
because we measure the zero position at the top and it slides down.
At the same time, the length of the string is increased according to or is equal to zero point one
to meters, while the angular velocity is given by Theta Dot, which is seven radians per second.
So you can see in the question I give you theatergoer directly.
I do not say theta is equal to seven D, I say theta that is seven.
I give you the angular acceleration immediately.
So the question is determine the forces experienced by the ball in the theater direction, the radial
direction and the direction at the instance at the instant when D three seconds.
So let us have a look.
We have formulas for the acceleration in each one of these directions.
OK, we know that the acceleration of the radial direction is all double minus or theatergoer squared
and that the acceleration in the theater direction is or doubled that plus two or three to dot, and
that the acceleration in the Z direction is a double dot.
And we have been given.
Functions of time for each one of these coordinates.
We've been told that the position in the Z direction is given by minus two to that, that our position
is given by zero comma one T and that theta dot, that angular velocity is given by seven radians per
second.
So let's apply that in our formulas.
The sum of the forces in the right direction is the mass multiplied by the acceleration in the right
direction.
That is the mass multiplied by our double dot minus or theta that squared, which is the mass multiplied
by now.
We have been given all.
And if we take the second derivative of or we find that it is zero minus or is zero point one dy multiplied
by theta that square we've been given theatergoer seven.
And if we plug in to be three seconds, we get a force of minus zero eight two, three, two Newtons.
Now we have implicitly assumed.
That the outward direction is positive.
Because I've said in the question that the length of the string is increased according to or is equal
to zero comma one T. So an increase in length is positive.
Right.
And so positive is outward.
And therefore, because we get a negative answer for the fourth, it means the forces in the direction
of the poll, the force is inward, which makes sense because that is what keeps the the ball from from
flinging out.
Then we look at the some of the forces in the theater direction, that's mass multiplied by the acceleration
of the theater direction, that's m multiplied by our theater.
Double plus two are not theater that that is M multiplied by or a zero one T theater double that.
We've been given theater dot as seven radiance per second.
That's a constant.
So Tom derivative of DataDot will be zero plus two.
Tom, derivative of all that zero point one multiplied by we've been given theatergoer as seven, and
if you substitute in 340 you get zero zero seven eight for Newton in the theater direction.
Now, the sum of the forces in this direction is multiplied by A-Z, and we know that the acceleration
of the direction, that's the second derivative of the Z position.
We have been told in the question that the ball slides down the pole according to minus zero to T..
So the first derivative of that.
Would be minus zero two and the second derivative would be zero.
So we see that there is no force in the Z direction.
Now, be careful how you interpret this, the ball slides down so clearly something is working in on
the ball, but there is no acceleration in this direction.
The sum of the forces in this direction is zero because, of course, we have the force of gravity working
down, but the friction.
Where the rope is attached to the pole is working up and therefore we only have a constant velocity
going in the same direction, meaning the acceleration in the same direction is zero.
And if the acceleration in a certain direction is zero, then we know the sum of the forces in that
direction is zero.
So there is no need to force acting on the ball in the direction that is what this result implies,
and if you read the question, you could see that because it said that such that its position in the
Z direction is given by minus zero comma to T. So it means that the position is a linear function of
T, that means the velocity will be constant.
That means the acceleration will be zero in the direction, meaning there is also no net force in this
direction.
So that is why this result of zero neutral force in the Z direction.
Makes sense, and if I say zero and force, I mean resultant force, the sum of the forces in this direction
is zero.
So this.
Example plus, the previous video serves to show you the basic principles of Newton's second law in
cylindrical code, and it's now not that no matter in which coordinate system we apply Newton's second
law, it does not change the law.
It's just that if we applied in a different coordinate system, we use different parameters to describe
the position of the particle and therefore we also use different parameters to describe the velocity
and then the acceleration of the particle.
That is why the formulas for acceleration look different in each one of the coordinate systems.
If we use rectangle coordinates, we would have.
Found that the same forces work in on the status for.
But it would have just been a long, different coordinate axis, because that is how we would have set
up the coordinate system.
So I just want to stress that Newton's second law remains universal in this case, and it's just the
coordinate systems that change.