Why is the packing fraction important for transporting watermelons and oranges?

[aspodkfpo]

Homework Statement

https://www.asi.edu.au/wp-content/uploads/2015/03/PhysicsASOE2014solutions.pdf
pg 16
Q13 c) ii) iii)

Relevant Equations

v=4pi r^3

https://www.asi.edu.au/wp-content/uploads/2015/03/PhysicsASOE2014solutions.pdf
Do not understand the q 13 c) ii), iii) logic at all.

 

[BvU]

The chocolates push each other out of the funnel.

 

[aspodkfpo]

The chocolates push each other out of the funnel.

ii) there should be a horizontal speed as well, and why would the chocolate pushed out be at the top and come down , i.e. 2r.

iii) understand this now

 

[Lnewqban]

ii) there should be a horizontal speed as well, and why would the chocolate pushed out be at the top and come down , i.e. 2r.

Each spherical chocolate within the outlet horizontal tube is pushed by the one inmediately upstream and it pushes the one inmediately downstream.
There is no gap among them.
Because of that, any fixed point of the outlet tube "sees" five spheres to go by in one second.
A row of five spheres has a length of five diameters.

$Speed=Distance~travelled~by~one~sphere/time=(2r)(5)/1=10r/s$

"As the level in the top of the funnel is steady the number of chocolates going in the top must be the same as the number coming out the bottom of the funnel. This means that 5 chocolates per sec- ond will come out of the funnels."

 

[aspodkfpo]

Homework Statement:: https://www.asi.edu.au/wp-content/uploads/2015/03/PhysicsASOE2014solutions.pdf
pg 16
Q13 c) ii) iii)
Relevant Equations:: v=4pi r^3

https://www.asi.edu.au/wp-content/uploads/2015/03/PhysicsASOE2014solutions.pdf
Do not understand the q 13 c) ii), iii) logic at all.

For c) iii), why did they add a /f for volume?

 

[jbriggs444]

For c) iii), why did they add a /f for volume?

I believe that is the packing fraction -- the total of the individual candy volumes divided by the bulk volume they occupy. Note the comment below: "b) some students had difficulty interpreting the meaning of a packing fraction"

One way of reasoning your way to the formula above is to start by asking: How much bulk volume is occupied by five candies?

The answer is: $5\frac{4 \pi r^3}{3f}$. They would occupy $5 \frac{4}{3}\pi r^3$ but they are inefficiently packed. So they occupy more volume than that.

 

[Lnewqban]

For c) iii), why did they add a /f for volume?

If you do a dimmensional analysis, you could see that the equation is for velocity rather than volume (Volume occupied by 5 spheres per second / Cross area of funnel).

"The packing fraction (f) is defined to be the ratio of the volume of some objects divided by the total volume of space which they occupy.
When arranged randomly the chocolates have a packing fraction of 0.64."

That is the reason that increases the cost of transporting watermelons respect to transporting oranges: you can pack less units in same truck.