2. In this problem, we will investigate some mechanical andelectrical properties of a frustum of a cone. It is a solidgenerated by the equation y = mx + c around the x-axis as shown inthe following figure.
All the relevant geometric parameters are given in thefigure.
(a) Find the arc length of this curve.
(b) Find the surface area of revolution of this equation whenit’s rotated around the x-axis.
(c) Find the volume of revolution for this curve once it'srotated about the x-axis.
(d) If the object has a material density of ρ, find the mass Mof it where M = volume × Density
(e) Assuming the solid generated by revolving the curve has auniform density ρ, find the mass of this object via integration andverify that the result you got in the previous part match this.Note that the mass is given by M = R ρ dV.
(f) Now let's say the density is not uniform and rather ρ(x) =ln(x) + 2, find the mass of this object again.
(g) Compute the moment of inertia I of this object which isgiven by I = R r2 dm
(h) This solid holds electrical resistance R and resistivity ϕ(which is a constant). Any solid with length L and cross-section Awill have R = ϕ L A . Using this information show that theresistance of the object in the figure is R = (ϕ(a − b)) /πr1 r2
- 2 In This Problem We Will Investigate Some Mechanical And Electrical Properties Of A Frustum Of A Cone It Is A Solid 1 (86.65 KiB) Viewed 44 times
2. In this problem, we will investigate some mechanical and electrical properties of frustum of a cone. It is a solid generated by the equation y=mx+c around the x axis as shown in the following figure. All the relevant geometric parameters are given in the figure. (a) Find the arc length of this curve. (b) Find the surface area of revolution of this equation when it's rotated around the x-axis. (c) Find the volume of revolution for this curve once its rotated about the x-axis. (d) If the object has a material density of ρ, find the mass M of it where M= Volume × Density (e) Assuming the solid generated by revolving the curve has a uniform density ρ, find the mass of this object via integration and verify that the result you got in the previous part match with this. Note that the mass is given by M=∫ρdV. (f) Now lets say the density is not uniform and rather ρ(x)=ln(x)+2, find the mass of this object again. (g) Compute the moment of inertia I of this object which is given by I=∫r2dm (h) This solid holds electrical resistance R and resitivity ϕ (which is a constant). Any solid with length L and cross section A will have R=ϕAL. Using these information show that the resistance of the object in the figure is R=πr1r2ϕ(a−b)