Uniform Solid Cylinder Example 10.8 A uniform solid cylinder has a radius R, mas
ID: 1553357 • Letter: U
Question
Uniform Solid Cylinder Example 10.8 A uniform solid cylinder has a radius R, mass M, and length L. Calculate its moment of inertia about its central axis (the z axis in the figure) SOLVE IT Conceptualize To simulate this situation imagine twirling a can of frozen juice around its central axis. Don't twirl a nonfrozen can of vegetable soup; it is not a rigid object! The liquid is able to move relative to the metal can Categorize This example is a substitution problem, using the definition of moment of inertia. We must reduce the integrand to a single variable. It is convenient to divide the cylinder into many cylindrical shells, each having radius r, thickness dr, and length L as shown in the figure. The density of the cylinder is p. The volume dV of each shell is its cross- sectional area multiplied by its length: dV J L dA L(2Tr dr. Express dm in terms of dr: Substitute this expression into the r2 dm equation for the moment of inertia of a rigid object: Use the total volume R2L of the cylinder to express its density dr Calculating I about the z axis for a uniform solid cylinder.Explanation / Answer
a) about cnetral axis I = 1/2mr^2
I = 1/2*density*volume*(1/2R^2) = 1/8*(m/R^3)*(R/2)^3*R^2 = 1/64mR^2 = 4.52 kg m^2
b) I tot = 1/2*185*1.25^2 = 144.53 kg m^2
remaining I = 144.53-4.52 = 140.01 kg m^2
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