Particle moving on an open cylinder

[nabeel17]

1. A force F causes a particle of mass m to move on the surface of an open
cylinder of radius R =
√x^2+y^2+z^2

(a) Write out the equations of motion of the particle in suitable coordinates.
(b) Construct a force F that causes the particle to perform simple harmonic motion
on the cylinder surface along the z-direction

I'm a little confused as to what the trajectory of the particle is...I'm assuming it is moving in circular motion around the cylinder but I'm not sure what the of the relevance of the open cylinder then.

If it is moving in circular motion, I was thinking the acceleration is just v^2/r perpendicular to the velocity (-r direction) but then since it is on a cylinder that would mean it wouldn't even move right?

 

[haruspex]

1. A force F causes a particle of mass m to move on the surface of an open cylinder of radius R = √x^2+y^2+z^2

That's not the equation of a cylinder, whichever way I parenthesise it.
The question is asking you to write down a general equation of motion for the particle given some arbitrary applied force F (=F(t) probably) and the fact that it is constrained to move on the surface of the cylinder. It's not entirely clear to me whether you are supposed to assume that F happens to have the property that it makes the particle stay in that cylindrical surface, or whether you are supposed to assume that F is completely arbitrary and the cylinder itself applies a normal force which constrains the motion, but it comes to the same thing in the end.

 

[Simon Bridge]

If I read R = √x^2+y^2+z^2 to mean $R=\sqrt{x^2+y^2+z^2}$ then that would be a sphere rather than a cylinder.

What the description is saying is that the particle is constrained to move on a particular shape. For a cylinder, radius R, centered on the z axis, this means the particle may have any value z coordinate but it's x and y coordinates must be related by $R^2=x^2+y^2$.

Part a does not require a trajectory - but an equation of motion.
I suspect this is part of a section on Lagrangian mechanics?
Whatever: you want to start out by selecting a suitable coordinate system.

 

[nabeel17]

sorry for late reply, Yes the radius is given by r=(x^2+y^2+z^2)^1/2 but I am still unclear of the trajectory. I will ask my professor for some more clarity :s

 

[Simon Bridge]

Yes the radius is given by r=(x^2+y^2+z^2)^1/2

Then it is not a cylinder. It is a sphere.

No trajectory is required.
Imagine the particle is a small ball bearing trapped between two concentric crystal spheres separated by the diameter of the ball and the whole in free fall. The ball is constrained to move on the surface of a sphere.

Add gravity, and the ball would have SHM.