Answer:
b) dm = ρab R dθ
c) m = ∫ ρab R dθ, from θ = 0 to θ = 2π
d) m = 2πρabR
Explanation:
b) We want to find the mass dm of a small chunk of the ring.
Mass is density times volume:
dm = ρ dV
Since a << R, we can approximate the volume as a rolled rectangular prism. Therefore, the volume of the chunk is the area of the cross section times the arc length.
dV = ab R dθ
dm = ρab R dθ
c) The mass of the entire ring is the sum of the masses of all the chunks.
m = ∫ dm
m = ∫ ρab R dθ, from θ = 0 to θ = 2π
d) ρ, a, b, and R are constants, so:
m = ρabR ∫ dθ
Evaluating the integral:
m = ρabR (θ|0 to 2π)
m = ρabR (2π − 0)
m = 2πρabR
