The shortest distance along cone

[abcd999]

Homework Statement

Obtain an integral formula for the length of a curve p(theta) along a right cone. use spherical coordinates p and theta.
Answer: L = integral from -pi/2 to pi/2 of sqrt(p'^2 + p^2/R^2)d(theta)

Homework Equations

p is distance from origin
altitude a, radius 1, cone's point is at origin, R=sqrt(a^2+1)

The Attempt at a Solution

I know ds^2 = dp^2 +sin^2p d(theta)^2 on spherical coordinates, but don't know how to begin translating that into the answer above

 

[mighty2000]

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Hello, thank you for your question. To obtain an integral formula for the length of a curve along a right cone, we can use the arc length formula:

L = integral from a to b of sqrt(1 + (dy/dx)^2)dx

In this case, we can let p(theta) be the curve on the cone, where p is the distance from the origin and theta is the angle around the cone. Using spherical coordinates, we can express this curve as p(theta) = p(theta)sin(theta).

Next, we can use the Pythagorean theorem to find the length of the curve at a specific point:

ds^2 = dp^2 + p^2d(theta)^2

Since we are on a right cone, we can use the altitude and radius given to find R, the distance from the origin to the curve at a specific point:

R = sqrt(a^2 + 1)

Substituting this into the arc length formula, we get:

L = integral from -pi/2 to pi/2 of sqrt(dp^2 + (p^2/R^2)d(theta)^2)

Simplifying this, we get:

L = integral from -pi/2 to pi/2 of sqrt(p'^2 + p^2/R^2)d(theta)

I hope this helps! Let me know if you have any further questions.