Acceleration of a bead sliding down a Helix

[physicsfun]

Can a bead desecnd down a helix have zero vertical component for its acceleration? (i.e. If I found my result to only has coefficients for its horizontal radial component unit vector (e_r), but zero for that of the vertical unit vector (e_z) ! ... That means I got zero vertical component for its acceleration right? Otherwise, please define the unit vectors <e_r, e_phi, e_z> for me please...?)
Is it possible for the bead to still have its descend down a vertical helix if it has zero component for its e_z unit vector's direction? Note that it is an HELIX... so perhaps the radial component alone, something like centripetal accel, or something like that, can pull it down to allow its downward descend still?? or no...? Why?

(See below for the question & how I tried to approach it) :

QUESTION:

A ball is constrained to slide down a frictionless helix path, with the helix axis to be vertical. The radius of the helix is R and each of the successive circular motion down the helix (distance of vertical drop after going round one full circular motion), is, 2pi*b.

The helix can be described by cylindrical coordinates: r=R, z=b*phi.

a) Write out the position vector and velocity vector.

b) Assuming that the bead is released from rest and allowed to determine freely along the helix, using convervation of energy to determine the time dependence of the angle theta and the vertical component of accelration.

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MY ATTEMPT:

I wrote out what represents the position and velocity vector, then used the conservation of energy equation, and integration to find angle phi, as shown below:

First, note unit vectors = <e_r, e_phi , e_s>, and d(e_r)/dt=e_phi*d(e_phi)/dt, and also d(e_phi)/dt = - [e_r*d(e_phi)/dt].

We found velocity vector to be v= R*(d-phi/dt)*e_r + b*(d-phi/dt)*e_z.

Thus, we also know that: acceleration vector = -R*(d-phi/d)*e_r + b*(d-phi^2/dt^2)*e_z

Now, using the conservation of energy equation of a ball falling from rest,
change of KE = change of PE, we get:
v=sqroot(2g*2pi*b).

Substituting the speed found from velocity vector: v=(d-phi/dt)*(R^2+b^2)^(1/2),
I get d-phi/dt= a bunch of constants.

So separating variables and integrating the resulting expression, we get phi:

phi=(sq.root[(4*pi*g*b/(R^2+b^2) ]*t ()

Taking the derivatives of phi twice with respect to time t, (which fits into the acceleration vector e_z coefficient component), however,
gives us zero.
i.e. d^2(phi)/dt^2 = 0

Can this be right??

i.e. The acceleration vector= a= -R*((d-phi/dr)^2)*e_r + b*0*e_z = -R*((d-phi/dr)^2)*e_r ... only!

i.e. the accel vector only have its coefficient for the radial unit vector (e_r), but ZERO coefficient for its vertical unit vector (e_z).

Is this possible??

Does my result mean the vertical component of acceleration is zero? (What are the definitions for the unit vector e_r, e_z etc?) Is there something wrong with my set up? How then should I do it?

Please comment on the approach i used, whether i have missed some concepts, and suggest an alternative better approach. Thank you!

 

[Thomas2]

You shouldn't be too confused by the helix shape. All that matters is the inclination of the helix thread i.e. the 'pitch angle'. As the latter is constant along the helix it should therefore be sufficient to consider a bead constricted to a straight line with the same inclination angle.

The pitch angle of the helix is

alpha = arctan[b/(2*PI*r)] ,

where b is the pitch (vertical drop between full circles) and r the radius of the helix.


The acceleration of the bead along the helix is thus

a = g*sin[alpha] ,

with g the gravitational acceleration.


Projecting this again back onto the vertical gives the vertical acceleration

av = g*sin[alpha] *sin[alpha] = g * sin^2[alpha]



With

z= b*phi(t) = 1/2*av*t^2

you have therefore the time dependence of the angle

phi(t) = 1/2/b *av *t^2 .

 

[charng]

Yes, it is possible for the bead to have zero vertical component for its acceleration while descending down a helix. This is because the motion of the bead along the helix is constrained to the path of the helix, which means that the radial component of acceleration (centripetal acceleration) is responsible for keeping the bead on the helix. The vertical component of acceleration does not contribute to this constraint and therefore can be zero.

To clarify, the unit vectors <e_r, e_phi, e_z> represent the directions of the three coordinate axes in cylindrical coordinates. e_r points in the direction of the radius of the helix, e_phi points in the direction of the tangent to the helix, and e_z points in the direction of the helix axis. These unit vectors are orthogonal to each other and are used to describe the position, velocity, and acceleration of the bead along the helix.

Your approach to solving the problem using the conservation of energy is correct. However, it is important to note that the conservation of energy equation only applies to the total energy of the system, not just the kinetic energy. This means that the potential energy (due to the height of the bead on the helix) must also be taken into account.

To find the time dependence of the angle theta and the vertical component of acceleration, you can use the fact that the bead is constrained to move along the helix. This means that the radial component of acceleration must be equal to the centripetal acceleration, which can be calculated using the velocity and radius of the helix. By setting these two components of acceleration equal to each other and solving for theta, you can find the time dependence of the angle theta. Similarly, you can use this relationship to find the time dependence of the vertical component of acceleration.

In summary, it is possible for the bead to have zero vertical component for its acceleration while descending down a helix due to the constraint of the motion along the helix. The unit vectors <e_r, e_phi, e_z> represent the directions of the cylindrical coordinate system. Your approach to solving the problem using the conservation of energy is correct, but it is important to consider the potential energy in addition to the kinetic energy. To find the time dependence of theta and the vertical component of acceleration, you can use the constraint of the motion along the helix. I hope this helps clarify your understanding of the problem.