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A circular ring is made of a material with uni-form density p (kg/m³). The ring has a rectangular cross section, as shown in the figure. The outer radius of the

ring is R. We would like to calculate the total mass of the ring. First, we will use the approximation that the width of the ring, a, is much, much smaller than the outer radius of the ring R (a « R). (a) On the top view figure, label a small “chunk" of the ring that subtends an arc d0. Note that in the a«R limit, the inner and outer arcs of your chunk have the same length. (b) Find an expression for the volume of the small chunk dV in terms of the variables in the figures and d0. (c) Find an expression for the mass of the small chunk dm in terms of p and dV. (d) Express the mass of the entire ring single integral in the angle 0. (Don't forget limits)as a (e) Evaluate the integral to find the total mass of the ring in the a < R limit.

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