1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: How the surface area of a cone changes; special relativity

  1. Jul 3, 2017 #1
    1. The problem statement, all variables and given/known data
    A cone has half angle θ0 and lateral surface area S0 in the frame in which the cone is at rest. If someone moves at relative speed β=v/c along the cones symmetry axis, what surface area will they see for the cone?

    2. Relevant equations
    I believe the lateral surface area of a cone is S=πh2tanθ, where θ is the half angle and h is the height of the cone.
    Length contraction is also relevant; h1=h0√(1-β2)

    3. The attempt at a solution
    S0 = πh02tanθ0
    S1 = πh12tanθ1

    If the cone had maximum radius R in the rest frame, it should remain R in the frame moving along the symmetry axis (because R is measured normal to this axis).
    So we have tanθ0=R/h0, and also tanθ1=R/h1, so we see tanθ1=(h0/h1)tanθ0

    Putting all this together (with the length contraction relation) it seems to be that S1=S0√(1-β2)

    However my book claims S1=S0√(1-(βcosθ)2); so where is my mistake?
  2. jcsd
  3. Jul 3, 2017 #2


    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Gold Member
    2017 Award

  4. Jul 3, 2017 #3
    Ahh. For some reason, when I thought over that formula, I had in my mind that "R=h*sinθ" instead of "R=h*tanθ"
    (I even remember thinking this formula over a couple times, and each time I kept incorrectly saying this to myself! I am tired.)

    So then the correct formula is actually S=πh2tanθ/cosθ.

    So then my answer is off by (needs to be multiplied by) a factor of cosθ0/cosθ1

    If you take the ratio of these two equations
    and simplify a fair amount (then use R/h0=tanθ as well as 1/(1+tan2θ)=cos2θ) then multiply it by my incorrect answer then it indeed becomes the correct answer, so that was the only mistake.

    Thank you Orodruin
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted