# Tangential movement in a waterslide

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1. Jul 9, 2017

### Fibo112

1. The problem statement, all variables and given/known data
A person enters a curve of a waterslide with velocity v. The curve is fully horizontal and has the shape of a circle segment of radius R, there is no friction and width of the curve can be neglected for calculating the centripedal force. The waterslide itself can be described as a semi circle with radius r. The question is to describe the tangential equation of motion for the angle a the person in the slide. We must than linearise this equation of motion with the small angle approx for sin and cos.

2. Relevant equations
F(centripedal)=mv^2/r

3. The attempt at a solution
The only forces acting are g and the normal force. Since the person is travelling in a circle of radius R with speed v there must be a centripedal force of mv^2/R (problem: the motion is not really circular since the person also has tangential movement in the slide). This centripedal force is horizontal so it cannot stem from gm -->it must come from the normal force. The horizontal component of the normal force is sin(a) N= aN=mv^2/R. The remaining force acting on the person is the vertical componenet of N and g =( mv^2/(Ra))-gm. The tangential component of this force is (mv^2/Ra)-g)sin a=(mv^2/Ra)-gm)a). This component is equal to a''mr. This gives a''=v^2/Rr-ga/r.

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2. Jul 9, 2017

### Hiero

It is not quite as simple as circular motion. (This equation isn't immediately useful if we don't know the radius of curvature of the motion.)

I would like to guide you based off your attempt, but I don't think I understand it. You seem to be taking the tangential component of the vertical component of the net force and equating it with mra'' ... Did you mean to take the tangential-component of the entire force instead of just a component of a component?

Assuming that is what you meant, it is still wrong to equate this with mra'' ... That would be the tangent-acceleration if the mass were oscillating in the slide (r) but not moving around the curve (R) ... but when it is also moving around the curve, things are not the same.

A safe approach would be to write the position of the mass in terms of some (inertial) coordinates and differentiate twice in time to find the acceleration.
An example of what I mean;
Position during circular motion can be described by the coordinates as R=<Rcos(θ), Rsin(θ)> so when we differentiate twice we get,
R''=<-θ'2Rcos(θ), -θ'2Rsin(θ)> = -θ'2R = -ω2R, the familiar centripetal-acceleration result.

3. Jul 9, 2017

### Fibo112

Thanks for the reply, I'm still stuck though :/

4. Jul 9, 2017

### Hiero

You want the general equations of motion for when the angle a can change in time right?

The particle is restricted to move on (part of) the surface of a torus (donut shape). I am suggesting you describe the cartesian coordinates of a point on this surface in terms of two angles (a and θ) analogous to my example:
If you paramerterize it correctly then finding equations of motion should just be a matter of carefully differentiating (and recognizing how v relates to θ').

If you don't understand what I am trying to say still then I apologize, but you'll have to explain your attempts more clearly if I am to help you more.

5. Jul 9, 2017

### haruspex

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You need another relationship between v and a. What's conserved?

6. Jul 9, 2017

### haruspex

You seem to be taking a as angle to the vertical (consistently). To use the small angle approximations you need it to be a deviation from an average angle.

7. Jul 9, 2017

### Fibo112

If the mechanical energy is conserved I would have another relationship since I know the relationship between the potential energy and the angle a. Is the mechanical energy conserved though? It would be if the normal force was always perpendicular to v..

8. Jul 9, 2017

### haruspex

Does the normal force do work?

9. Jul 9, 2017

### Fibo112

With circular motion in a plane I know it doesnt since it is perpendicular to v, in 3d I'm not sure

10. Jul 9, 2017

### haruspex

A force only does work if there is motion parallel to it. I think you are getting confused with the constant speed condition.

11. Jul 9, 2017

### Fibo112

Yeah thats why I said it doesnt do work since it is perpendicular?

12. Jul 9, 2017

### haruspex

Yes, sorry. Anyway, work is conserved here.

13. Jul 9, 2017

### Fibo112

Now I have this relationship.

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14. Jul 9, 2017

### haruspex

Sin(a) for PE?

15. Jul 9, 2017

### Fibo112

I thought sin(a)r is the height and then I just multiply that with mg?

16. Jul 9, 2017

### haruspex

Think again. E.g. consider 45 degrees.

17. Jul 9, 2017

### Fibo112

Ups, PE= r(1-cos a)mg

18. Jul 9, 2017

### Fibo112

I still don't know how to continue though...

19. Jul 9, 2017

### Fibo112

At this point I would really appreciate it if someone could just give me a solution...

20. Jul 9, 2017

### haruspex

Don't differentiate. Just use the energy expression to substitute for v in your acceleration equation.
Not allowed on homework forums.
As I noted, you cannot use the small angle approximation for a. It will not generally be that small. Instead, find the equilibrium value for a and make the small angle approximation for deviations from there.