Computing mass with a denstiy function

In summary: So the mass is \pi/2.In summary, to find the total mass of a wire bent in a quarter circle with given parametric equations and density function, you can use the formula for the differential of length and integrate the density function along the curve. However, in this specific case, the density is constant and the mass is simply 1/4 the circumference of a circle of radius 1, giving a total mass of \pi/2.
  • #1
MasterWu77
21
0

Homework Statement


Compute the total mass of a wire bent in a quarter circle with parametric equations: x=cos(t), y=sin(t), 0[tex]\leq[/tex] t [tex]\leq[/tex] [tex]\pi[/tex]/2
and density function [tex]\rho[/tex](x,y) = x^2+y^2


Homework Equations



not exactly too sure which equations if any i need to use. maybe the jacobian

The Attempt at a Solution



i simply substituted the x and y into the density function to get

(6cost)^2 + (6sint)^2 and took the integral of that with the bounds of integration from 0 to pi/2. the answer i am getting is 56.549 and is wrong and I'm not sure if there's an extra step i need to do
 
Physics news on Phys.org
  • #2
You want, of course, to integrate the density function along the length of the given curve. The differential of length, when the curve is given by parametric equations, x= x(t), y= y(t), is
[tex]\sqrt{\left(\frac{dx}{dt}\right)^2+ \left(\frac{dy}{dx}\right)^2}dt[/tex].

However, here, you should be able to see that the density at point (x,y) is [itex]cos^2(t)+ sin^2(t)= 1[/itex]. (I have no idea where you got the "6" in your formula). Since that density is constant around the arc, the mass is just 1 times the length of the curve. And that is simply 1/4 the circumference of a circle of radius 1.
 

Related to Computing mass with a denstiy function

1. How is density related to mass?

Density is the measure of mass per unit volume. In other words, it is the amount of mass in a given volume of a substance. The higher the density, the more mass is contained in a specific volume.

2. What is the formula for computing mass with a density function?

The formula for computing mass with a density function is mass = density x volume. This means that to find the mass of an object, you multiply the density of the substance by the volume it occupies.

3. How is density measured?

Density is typically measured in units of grams per cubic centimeter (g/cm³) or kilograms per cubic meter (kg/m³). It can be calculated by dividing the mass of a substance by its volume.

4. Can the density of a substance change?

Yes, the density of a substance can change depending on factors such as temperature, pressure, and composition. For example, water has a higher density when it is colder and a lower density when it is warmer.

5. How is computing mass with a density function useful?

Computing mass with a density function is useful in many scientific and engineering applications. It allows us to determine the mass of a substance without physically weighing it, which can be difficult or even impossible in some cases. It also helps in understanding the properties and behavior of different materials.

Similar threads

Replies
10
Views
1K
Replies
1
Views
706
Replies
1
Views
1K
Replies
8
Views
1K
Replies
4
Views
1K
Replies
3
Views
1K
Replies
3
Views
1K
Back
Top