What is the velocity of a falling object with air resistance?

[Antony Jose]

Homework Statement

I am trying to develop simulation for a falling object subject to air resistance. Object is similar to Samara seed. object is considered to be under steady vertical descend.
know variable :
surface area of object (A)
weight of object (W)
c_d: drag coefficient
object rotates while falling.
length from center of rotation to tip of the object (L)
inclines at an angle beta (β)
Swept disk radius (r) = L*cos(β)

Homework Equations

terminal velocity(V_s)= √[(2*W)/(c_d*ρ*A)
Disk loading= W/A=ρ(V_s²-V_f²)/2

The Attempt at a Solution

I am stuck in this position and don't know how to approach the problem from here. want to know how to calculate the velocity of object and angular velocity of the tip of the object.

 

[CWatters]

I can't help with the aerodynamics but...

object is considered to be under steady vertical descend.

If it's descending at constant velocity then at that velocity the vertical force upwards (caused by a combination of lift and drag) is equal to the force due to gravity downwards.

 

[CWatters]

Google suggests the aerodynamics is quite complicated and possibly not fully understood..

http://mappingignorance.org/2013/05/15/autorotating-seeds-to-fly-or-to-die/

 

[Antony Jose]

thanks for the replies.
I had some more doubt. Like how to draw the helix created by the object if it start rotating 1 meter above the ground.
given that we know the velocity of fall, angular velocity of rotation and radius of helix.

 

[CWatters]

If we assume that the seed very quickly reaches terminal velocity then that's solvable...

The vertical velocity (v) and the height (h) give you the time it takes to fall (t)...

t = h/v

Then the time (t) and angular velocity (ω) gives you the number of revolutions (n) it makes before hitting the ground..

n = ωt/2π (Note I had to edit this line)

Then the height (h) and the number of revolutions (n) gives you the pitch of the helix (p) in meters (p = the distance it falls per revolution)..

p = h/n

I'm not sure how you define the radius of the helix? I think seeds with a single wing rotate about a point on the wing near the seed head. So the seed and wing tip move in a helix (different radii) but the centre of rotation descends in a straight line (ignoring wind).

 

[Antony Jose]

If we assume that the seed very quickly reaches terminal velocity then that's solvable...

The vertical velocity (v) and the height (h) give you the time it takes to fall (t)...

t = h/v

Then the time (t) and angular velocity (ω) gives you the number of revolutions (n) it makes before hitting the ground..

n = ωt/2π (Note I had to edit this line)

Then the height (h) and the number of revolutions (n) gives you the pitch of the helix (p) in meters (p = the distance it falls per revolution)..

p = h/n

I'm not sure how you define the radius of the helix? I think seeds with a single wing rotate about a point on the wing near the seed head. So the seed and wing tip move in a helix (different radii) but the centre of rotation descends in a straight line (ignoring wind).

thank you for the help.
center of rotation can be found by find center of mass (C.M). Most of the seed have this as center of rotation.

 

[Antony Jose]

Can you help me with finding the angular velocity of the object.
I took angular velocity as terminal velocty (v_trmnl)*sin(Θ)/radius, since no other force is acting on the object. [Θ is the angle of attack]
But I think there is some thing wron.

 

[CWatters]

I took angular velocity as terminal velocty (vtrmnl)*sin(Θ)/radius,

I make it

Angular velocity (Rads/s) = V_trmnl / (Radius * Tan(Θ))

In one revolution it falls a distance h so

Tan(θ) = h/circumference
so
h = Tan(θ) * 2πR

The time for one revolution is t..

t = h/V_termnl
t = (Tan(θ) * 2πR)/V_termnl

Angular Velocity = angle/time
= 2π/t
= 2π/((Tan(θ) * 2πR)/V_termnl)
= V_termnl/(Radius * Tan(Θ))

 

[CWatters]

I should add that in my post above the angle θ is the pitch angle of the helix (not the aerodynamic angle of attack). I don't think you can easily calculate the angular velocity from the aerodynamic pitch.

 

[Antony Jose]

I should add that in my post above the angle θ is the pitch angle of the helix (not the aerodynamic angle of attack). I don't think you can easily calculate the angular velocity from the aerodynamic pitch.

How to find pitch angle?
I think we can find it using h and circumference. Using Pythagoras theorm.

 

[CWatters]

Correct although you don't need Pythagoras...

The pitch angle of the helix (distance descended per revolution) = Tan^-1(h/circumference)

But note that this is not the same as the aerodynamic pitch.

 

[Antony Jose]

Correct although you don't need Pythagoras...

The pitch angle of the helix (distance descended per revolution) = Tan^-1(h/circumference)

But note that this is not the same as the aerodynamic pitch.

problem in this is that i couldn't find 'h'

In one revolution it falls a distance h

.