Resistance of a cone section

In summary, we have a conically-shaped material in a circuit with varying resistivity and radius. The resistivity is given by rho=(6*10^6)*x^4 (where x is measured in meters and rho is measured in ohm*meters), and the radius ranges from r1=xinitial=3 cm to r2=xfinal=8.4 cm. To find the total resistance, we integrate (6/pi*10^6)*x^2 dx from xinitial to xfinal, taking into account that the cross-sectional area increases as a function of x.
  • #1
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Homework Statement


A piece of conically-shaped material is placed in a circuit along the x-axis. The resistivity of this material varies as rho=(6*10^6)*x^4 (where x is measured in meters and rho is measured in ohm*meters), and its radius varies linearly as a function of x, ranging from r1=xinitial=3 cm to r2=xfinal=8.4 cm.


Homework Equations


R=rho*L/A


The Attempt at a Solution



Well current is flowing through the cone section so the cross sectional area is a circle, with the radius increasing proportional to x, in fact the are equal.

So dR=rho*dx/(2*pi*r) and dx=dr and x=r so plugging in for r

dR=(6*10^6)*r^4/(2*pi*r) dr = (3/pi * 10^6)*r^3 dr

which integrating from 0.03 to 0.084 m i got 11.69 Ohm, doesn't seem quite right did I mess up somewhere?
 
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  • #2
You sure that ##A = 2 \pi r## ?
 
  • #3
Would it be something else? A bunch of little rings expanding out would make a circle yes?
 
  • #4
DrIxn said:
Would it be something else? A bunch of little rings expanding out would make a circle yes?

Shouldn't A be the cross-sectional area of the cone? It didn't appear that you were integrating in the radial direction...
 
  • #5
Okay so the cross sectional area would be a circle, A=pi*r^2 , and since r varies linearly with the length of the cone A=pi*x^2

And integrating with respect to x..

(6/pi*10^6)*x^2 dx from xi to xf?
 
  • #6
DrIxn said:
Okay so the cross sectional area would be a circle, A=pi*r^2 , and since r varies linearly with the length of the cone A=pi*x^2

And integrating with respect to x..

(6/pi*10^6)*x^2 dx from xi to xf?

Yup, that looks better.
 

1. What is the resistance of a cone section?

The resistance of a cone section is the force that opposes the motion of the cone as it moves through a fluid (such as air or water).

2. How is the resistance of a cone section calculated?

The resistance of a cone section can be calculated using the formula: R = 0.5 x ρ x v^2 x A x C, where ρ is the density of the fluid, v is the velocity of the cone, A is the cross-sectional area of the cone, and C is the drag coefficient.

3. What factors affect the resistance of a cone section?

The resistance of a cone section is affected by the shape of the cone, the velocity of the cone, the density of the fluid, and the drag coefficient. The shape of the cone and the drag coefficient are the most important factors, as they determine how streamlined the cone is and how much drag it will experience.

4. How does the resistance of a cone section change with velocity?

The resistance of a cone section increases with velocity, as the faster the cone moves through the fluid, the more drag it will experience. This is due to the increase in kinetic energy and pressure as the cone accelerates.

5. How does the resistance of a cone section compare to other shapes?

The resistance of a cone section is generally lower than that of other shapes, such as a flat plate or a sphere, as it has a more streamlined shape. However, the resistance can vary greatly depending on the specific dimensions and characteristics of the cone.

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