Calculate the terminal velocity for a pollen grain

In summary, the terminal velocity for a pollen grain falling through air can be calculated using the drag force equation. Assuming a drag coefficient of 0.5, a diameter of 7 µm, and a density of 0.3 g/cm3, the correct solution is found to be 0.145m/s. The mistake in the attempt at the solution was using the wrong formula for the volume of a sphere, resulting in an incorrect mass calculation. The correct formula is 4/3pi*r^3, not 4/3pi*r^2.
  • #1
sarmar
8
0

Homework Statement



Calculate the terminal velocity for a pollen grain falling through the air using the drag force equation. Assume the pollen grain has a diameter of 7 µm and a density of 0.3 g/cm3.


Homework Equations



Vterm= sq.rt of 2mg/pA
Volume= 4/3pi*r^2
Density = m/v

The Attempt at a Solution



I am given the answer but need to show how to get there.
Here is what I have, can someone point out where exactly I am going wrong?
Thank you!

Vterm= sq.rt of 2mg/pA

p= air density of 1.3kg and

A = cross section of the pollen grain
A= 7µm = 0.000007m

To find mass of pollen grain:
radius= 1/2diameter
= 1/2(0.000007)
= 0.0000035m
Volume= 4/3pi*r^2
= 4/3pi(0.0000035)^2
= 5.13x10^-11 m^3
Density = m/v
m=DV
=0.3g/m^2 (5.13x10^-11 m^3)
= 1.54 x10^-11g
to kg = 1.54 x10^-14 kg

So, Vterm= sq.rt of 2mg/pA
Vterm = sq.rt. of 2(1.54 x10^-14 kg)(9.8m/s^2) / (1.3kg/m^2)(0.000007m)
=0.0018m/s

But I know this is wrong since I am given the final answer, which is 0.145m/s
I've worked it out so many times and I'm stuck. Please help.
 
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  • #2
sarmar said:
Vterm= sq.rt of 2mg/pA
Are you assuming a drag coefficient = 1?

A = cross section of the pollen grain
A= 7µm = 0.000007m
7µm is the diameter of the pollen grain.
 
  • #3
Doc Al said:
Are you assuming a drag coefficient = 1?


7µm is the diameter of the pollen grain.


I am not given any information for drag coefficient. So I suppose I was assuming 1. I found online the drag coefficient of a spherical object is approx. 0.5

I thought the diameter was the same as the cross-section.
Area of a circle? A=p*r^2
=pi*0.0000035^2
=3.85x10^-11 so,
A=cross-section=3.85x10^-11

so then I just re-figure the mass ?
 
  • #4
sarmar said:
I am not given any information for drag coefficient. So I suppose I was assuming 1. I found online the drag coefficient of a spherical object is approx. 0.5
That sounds OK.
I thought the diameter was the same as the cross-section.
No, A is the cross-sectional area.
Area of a circle? A=p*r^2
=pi*0.0000035^2
=3.85x10^-11 so,
A=cross-section=3.85x10^-11
Looks OK.

so then I just re-figure the mass ?
Why would you re-figure the mass?
 
  • #5
Doc Al said:
Why would you re-figure the mass?


Well because I'm still not getting the right answer...
 
  • #6
sarmar said:
To find mass of pollen grain:
radius= 1/2diameter
= 1/2(0.000007)
= 0.0000035m
Volume= 4/3pi*r^2
That should be: 4/3pi*r^3.
 
  • #7
Thank you!
 

1. What is terminal velocity?

Terminal velocity is the maximum constant velocity that an object can reach when falling through a fluid, such as air or water. At terminal velocity, the forces of gravity and air resistance on the object are equal, resulting in a balanced net force and no further acceleration.

2. How is terminal velocity calculated?

To calculate terminal velocity for a pollen grain, you will need to use the following formula: Vt = sqrt((2mg)/(ρACd)), where Vt is the terminal velocity, m is the mass of the pollen grain, g is the acceleration due to gravity, ρ is the density of air, A is the projected area of the pollen grain, and Cd is the drag coefficient.

3. What is the drag coefficient?

The drag coefficient is a dimensionless value that represents the amount of air resistance or drag that an object experiences as it moves through a fluid. It is influenced by factors such as the shape, size, and surface properties of the object.

4. How does air density affect terminal velocity?

The density of air plays a significant role in determining the terminal velocity of an object. As air density increases, the drag force also increases, resulting in a lower terminal velocity. This means that terminal velocity will be lower at higher altitudes where air density is lower.

5. Are there any other factors that can affect the terminal velocity of a pollen grain?

Yes, there are other factors that can affect the terminal velocity of a pollen grain, such as the shape and size of the grain, air temperature and pressure, and the presence of other objects or obstacles in its path. These factors can alter the drag force and ultimately affect the terminal velocity of the pollen grain.

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