Physics Work Problem

Question

A tank in the shape of an inverted right circular cone has height 6 meters and radius 3 meters. It is filled with 5 meters of hot chocolate. Find the work required to empty the tank by pumping the hot chocolate over the top of the tank. The density of hot chocolate is 1070 kg.m^3. Your answer must include the correct units.

Answer 1

Let vertex of cone is origin and x-axis along cone-axis a directed up. Let we calculate elementary work dW

to muve elementary volume of hot chocolate getting by horizontal section through point with coordinate x on x-axis and elementary move by dx: radius r of section r/x = 3/6; r = x/2; so area of section πx²/4.

dW = ρgπx²/4(6 - x)dx; W = πρg/4 ∫₀⁵x²(6 - x)dx = πgρ/4(2x³ - x⁴/4)₀⁵ = π·9.81.482m/s²·1070/4kg/m³(250m⁴ - 156.25m⁴) = 772883.3345 J