A circular ring is made of a material with uni- form density p (kg/m). The ring...

  

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A circular ring is made of a material with uni- form density p (kg/m). The ring has a rectan- gular 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. b Q a a Cross-section view Top view R 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 de. 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 do. (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 as a single integral in the angle 0. (Don't forget limits) (e) Evaluate the integral to find the total mass of the ring in the a A circular ring is made of a material with uni- form density p (kg/m). The ring has a rectan- gular 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. b Q a a Cross-section view Top view R 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 de. 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 do. (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 as a single integral in the angle 0. (Don't forget limits) (e) Evaluate the integral to find the total mass of the ring in the a

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a On the top view figure label a small chunk of the ring that subtends an arc de Note that in the a ... View the full answer