Water flows in a shallow semicircular channel with inner and outer radii of 1 m and 2 m (see figure). At a point P(r, θ) in the channel, the flow is in the tangential direction (counterclockwise along circles), and it depends only on r, the distance from the center of the semicircles. 

a. Express the region formed by the channel as a set in polar coordinates.

b. Express the inflow and outflow regions of the channel as sets in polar coordinates.

c. Suppose the tangential velocity of the water in m>s is given by v(r) = 10r, for 1 ≤ r ≤ 2. Is the velocity greater at (1.5, π/4) or (1.2, 3π/4)? Explain.

d. Suppose the tangential velocity of the water is given by Is the velocity greater at (1.8, π/6) or (1.3, 2π/3)? Explain.

e. The total amount of water that flows through the channel (across a cross section of the channel θ = θ₀ ) is proportional toIs the total flow through the channel greater for the flow in part (c) or (d)?

Transcribed Image Text:

, for 1 s r s 2. v(r)